Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group {\Gamma} acting in a measure-preserving fashion on a probability space {(X,\mu)}, thus each group element {\gamma \in \Gamma} gives rise to a measure-preserving map {T^\gamma: X \rightarrow X}. Define the third Gowers-Host-Kra seminorm {\|f\|_{U^3(X)}} of a function {f \in L^\infty(X)} via the formula

\displaystyle  \|f\|_{U^3(X)}^8 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2,h_3 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2,\omega_3 \in \{0,1\}}

\displaystyle {\mathcal C}^{\omega_1+\omega_2+\omega_3} f(T^{\omega_1 h_1 + \omega_2 h_2 + \omega_3 h_3} x)\ d\mu(x)

where {\Phi_n} is a Folner sequence for {\Gamma} and {{\mathcal C}: z \mapsto \overline{z}} is the complex conjugation map. One can show that this limit exists and is independent of the choice of Folner sequence, and that the {\| \|_{U^3(X)}} seminorm is indeed a seminorm. A Conze-Lesigne system is an ergodic measure-preserving system in which the {U^3(X)} seminorm is in fact a norm, thus {\|f\|_{U^3(X)}>0} whenever {f \in L^\infty(X)} is non-zero. Informally, this means that when one considers a generic parallelepiped in a Conze–Lesigne system {X}, the location of any vertex of that parallelepiped is more or less determined by the location of the other seven vertices. These are the important systems to understand in order to study “complexity two” patterns, such as arithmetic progressions of length four. While not all systems {X} are Conze-Lesigne systems, it turns out that they always have a maximal factor {Z^2(X)} that is a Conze-Lesigne system, known as the Conze-Lesigne factor or the second Host-Kra-Ziegler factor of the system, and this factor controls all the complexity two recurrence properties of the system.

The analogous theory in complexity one is well understood. Here, one replaces the {U^3(X)} norm by the {U^2(X)} norm

\displaystyle  \|f\|_{U^2(X)}^4 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2 \in \{0,1\}} {\mathcal C}^{\omega_1+\omega_2+\omega_3} f(T^{\omega_1 h_1 + \omega_2 h_2} x)\ d\mu(x)

and the ergodic systems for which {U^2} is a norm are called Kronecker systems. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group {Z} equipped with Haar probability measure and a translation action {T^\gamma \colon z \mapsto z + \phi(\gamma)} for some homomorphism {\phi: \Gamma \rightarrow Z} with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).

We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) nilsystems, in which the space {X} is a quotient {G/\Lambda} of a two-step nilpotent Lie group {G} by a lattice {\Lambda} (equipped with Haar probability measure), and the action is given by a translation {T^\gamma x = \phi(\gamma) x} for some group homomorphism {\phi: \Gamma \rightarrow G}. For instance, the Heisenberg {{\bf Z}}-nilsystem

\displaystyle  \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix}

with a shift of the form

\displaystyle  Tx = \begin{pmatrix} 1 & \alpha & 0 \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end{pmatrix} x

for {\alpha,\beta} two real numbers with {1,\alpha,\beta} linearly independent over {{\bf Q}}, is a Conze-Lesigne system. As the base case of a well known result of Host and Kra, it is shown in fact that all Conze-Lesigne {{\bf Z}}-systems are inverse limits of nilsystems (previous results in this direction were obtained by Conze-Lesigne, Furstenberg-Weiss, and others). Similar results are known for {\Gamma}-systems when {\Gamma} is finitely generated, thanks to the thesis work of Griesmer (with further proofs by Gutman-Lian and Candela-Szegedy). However, this is not the case once {\Gamma} is not finitely generated; as a recent example of Shalom shows, Conze-Lesigne systems need not be the inverse limit of nilsystems in this case.

Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which {G} is a locally compact Polish group rather than a Lie group:

Theorem 1 (Classification of Conze-Lesigne systems) Let {\Gamma} be a countable abelian group, and {X} an ergodic measure-preserving {\Gamma}-system. Then {X} is a Conze-Lesigne system if and only if it is the inverse limit of translational systems {G/\Lambda}, where {G} is a nilpotent locally compact Polish group of nilpotency class two, and {\Lambda} is a lattice in {G} (and also a lattice in the commutator group {[G,G]}), with {G/\Lambda} equipped with the Haar probability measure and a translation action {T^\gamma x = \phi(\gamma) x} for some homomorphism {\phi: \Gamma \rightarrow G}.

In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm {U^3(G)} for an arbitrary finite abelian group {G} (with no restrictions on the order of {G}, in particular our result handles the case of even and odd {|G|} in a unified fashion). In principle, having a higher order version of this theorem will similarly allow us to derive inverse theorems for {U^{s+1}(G)} norms for arbitrary {s} and finite abelian {G}; we hope to investigate this further in future work.

We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension {Z \rtimes_\rho K}, where {Z} is a Kronecker system (a rotational system on a compact abelian group {Z = (Z,+)} and translation action {\phi: \Gamma \rightarrow Z}), {K = (K,+)} is another compact abelian group, and the cocycle {\rho = (\rho_\gamma)_{\gamma \in \Gamma}} is a collection of measurable maps {\rho_\gamma: Z \rightarrow K} obeying the cocycle equation

\displaystyle  \rho_{\gamma_1+\gamma_2}(x) = \rho_{\gamma_1}(T^{\gamma_2} x) + \rho_{\gamma_2}(x) \ \ \ \ \ (1)

for almost all {x \in Z}. Furthermore, {\rho} is of “type two”, which means in this concrete setting that it obeys an additional equation

\displaystyle  \rho_\gamma(x + z_1 + z_2) - \rho_\gamma(x+z_1) - \rho_\gamma(x+z_2) + \rho_\gamma(x) \ \ \ \ \ (2)

\displaystyle  = F(x + \phi(\gamma), z_1, z_2) - F(x,z_1,z_2)

for all {\gamma \in \Gamma} and almost all {x,z_1,z_2 \in Z}, and some measurable function {F: Z^3 \rightarrow K}; roughly speaking this asserts that {\phi_\gamma} is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where {Z} is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group {K} is a circle {{\bf T} = {\bf R}/{\bf Z}}, one can use some Fourier analysis to convert (2) into the more tractable Conze-Lesigne equation

\displaystyle  \rho_\gamma(x+z) - \rho_\gamma(x) = F_z(x+\phi(\gamma)) - F_z(x) + c_z(\gamma) \ \ \ \ \ (3)

for all {\gamma \in \Gamma}, all {z \in Z}, and almost all {x \in Z}, where for each {z}, {F_z: Z \rightarrow K} is a measurable function, and {c_z: \Gamma \rightarrow K} is a homomorphism. (For technical reasons it is often also convenient to enforce that {F_z, c_z} depend in a measurable fashion on {z}; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group {K} (as long as one has the measurability in {z} mentioned previously), but the converse turns out to fail for some groups {K}, such as solenoid groups (e.g., inverse limits of {{\bf R}/2^n{\bf Z}} as {n \rightarrow \infty}), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group {K = \frac{1}{N}{\bf Z}/{\bf Z}}. Putting together the {K={\bf T}} and {K = \frac{1}{N}{\bf Z}/{\bf Z}} cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian Lie groups {K} (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system {X} as a translational system {G/\Lambda}, where the Host-Kra group {G} is the set of all pairs {(z, F_z)} that solve an equation of the form (3) (with these pairs acting on {X \equiv Z \rtimes_\rho K} by the law {(z,F_z) \cdot (x,k) := (x+z, k+F_z(x))}), and {\Lambda} is the stabiliser of a point in this system. This then establishes the theorem in the case when {K} is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of {X} form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)

There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of {G} on {X} is technically only a near-action rather than a genuine action, and as such one cannot directly define {\Lambda} to be the stabiliser of a point without running into multiple problems. To fix this, one has to pass to a topological model of {X} in which the action becomes continuous, and the stabilizer becomes well defined, although one then has to work a little more to check that the action is still transitive. This can be done via Gelfand duality; we proceed using a mixture of a construction from this book of Host and Kra, and the machinery in this recent paper of Asgar and myself.

Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case {K = \frac{1}{N}{\bf Z}/{\bf Z}}. As this group embeds into the torus {{\bf T}}, it is easy to use existing methods obtain (3) but with the homomorphism {c_z} and the function {F_z} taking values in {{\bf R}/{\bf Z}} rather than in {\frac{1}{N}{\bf Z}/{\bf Z}}. The main task is then to fix up the homomorphism {c_z} so that it takes values in {\frac{1}{N}{\bf Z}/{\bf Z}}, that is to say that {Nc_z} vanishes. This only needs to be done locally near the origin, because the claim is easy when {z} lies in the dense subgroup {\phi(\Gamma)} of {Z}, and also because the claim can be shown to be additive in {z}. Near the origin one can leverage the Steinhaus lemma to make {c_z} depend linearly (or more precisely, homomorphically) on {z}, and because the cocycle {\rho} already takes values in {\frac{1}{N}{\bf Z}/{\bf Z}}, {N\rho} vanishes and {Nc_z} must be an eigenvalue of the system {Z}. But as {Z} was assumed to be separable, there are only countably many eigenvalues, and by another application of Steinhaus and linearity one can then make {Nc_z} vanish on an open neighborhood of the identity, giving the claim.

As math educators, we often wish out loud that our students were more excited about mathematics. I finally came across a video that indicates what such a world might be like:

A popular way to visualise relationships between some finite number of sets is via Venn diagrams, or more generally Euler diagrams. In these diagrams, a set is depicted as a two-dimensional shape such as a disk or a rectangle, and the various Boolean relationships between these sets (e.g., that one set is contained in another, or that the intersection of two of the sets is equal to a third) is represented by the Boolean algebra of these shapes; Venn diagrams correspond to the case where the sets are in “general position” in the sense that all non-trivial Boolean combinations of the sets are non-empty. For instance to depict the general situation of two sets {A,B} together with their intersection {A \cap B} and {A \cup B} one might use a Venn diagram such as

venn

(where we have given each region depicted a different color, and moved the edges of each region a little away from each other in order to make them all visible separately), but if one wanted to instead depict a situation in which the intersection {A \cap B} was empty, one could use an Euler diagram such as

euler

One can use the area of various regions in a Venn or Euler diagram as a heuristic proxy for the cardinality {|A|} (or measure {\mu(A)}) of the set {A} corresponding to such a region. For instance, the above Venn diagram can be used to intuitively justify the inclusion-exclusion formula

\displaystyle  |A \cup B| = |A| + |B| - |A \cap B|

for finite sets {A,B}, while the above Euler diagram similarly justifies the special case

\displaystyle  |A \cup B| = |A| + |B|

for finite disjoint sets {A,B}.

While Venn and Euler diagrams are traditionally two-dimensional in nature, there is nothing preventing one from using one-dimensional diagrams such as

venn1d

or even three-dimensional diagrams such as this one from Wikipedia:

venn-3d

Of course, in such cases one would use length or volume as a heuristic proxy for cardinality or measure, rather than area.

With the addition of arrows, Venn and Euler diagrams can also accommodate (to some extent) functions between sets. Here for instance is a depiction of a function {f: A \rightarrow B}, the image {f(A)} of that function, and the image {f(A')} of some subset {A'} of {A}:

afb

Here one can illustrate surjectivity of {f: A \rightarrow B} by having {f(A)} fill out all of {B}; one can similarly illustrate injectivity of {f} by giving {f(A)} exactly the same shape (or at least the same area) as {A}. So here for instance might be how one would illustrate an injective function {f: A \rightarrow B}:

afb-injective

Cartesian product operations can be incorporated into these diagrams by appropriate combinations of one-dimensional and two-dimensional diagrams. Here for instance is a diagram that illustrates the identity {(A \cup B) \times C = (A \times C) \cup (B \times C)}:

cartesian

In this blog post I would like to propose a similar family of diagrams to illustrate relationships between vector spaces (over a fixed base field {k}, such as the reals) or abelian groups, rather than sets. The categories of ({k}-)vector spaces and abelian groups are quite similar in many ways; the former consists of modules over a base field {k}, while the latter consists of modules over the integers {{\bf Z}}; also, both categories are basic examples of abelian categories. The notion of a dimension in a vector space is analogous in many ways to that of cardinality of a set; see this previous post for an instance of this analogy (in the context of Shannon entropy). (UPDATE: I have learned that an essentially identical notation has also been proposed in an unpublished manuscript of Ravi Vakil.)

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In everyday usage, we rely heavily on percentages to quantify probabilities and proportions: we might say that a prediction is {50\%} accurate or {80\%} accurate, that there is a {2\%} chance of dying from some disease, and so forth. However, for those without extensive mathematical training, it can sometimes be difficult to assess whether a given percentage amounts to a “good” or “bad” outcome, because this depends very much on the context of how the percentage is used. For instance:

  • (i) In a two-party election, an outcome of say {51\%} to {49\%} might be considered close, but {55\%} to {45\%} would probably be viewed as a convincing mandate, and {60\%} to {40\%} would likely be viewed as a landslide.
  • (ii) Similarly, if one were to poll an upcoming election, a poll of {51\%} to {49\%} would be too close to call, {55\%} to {45\%} would be an extremely favorable result for the candidate, and {60\%} to {40\%} would mean that it would be a major upset if the candidate lost the election.
  • (iii) On the other hand, a medical operation that only had a {51\%}, {55\%}, or {60\%} chance of success would be viewed as being incredibly risky, especially if failure meant death or permanent injury to the patient. Even an operation that was {90\%} or {95\%} likely to be non-fatal (i.e., a {10\%} or {5\%} chance of death) would not be conducted lightly.
  • (iv) A weather prediction of, say, {30\%} chance of rain during a vacation trip might be sufficient cause to pack an umbrella, even though it is more likely than not that rain would not occur. On the other hand, if the prediction was for an {80\%} chance of rain, and it ended up that the skies remained clear, this does not seriously damage the accuracy of the prediction – indeed, such an outcome would be expected in one out of every five such predictions.
  • (v) Even extremely tiny percentages of toxic chemicals in everyday products can be considered unacceptable. For instance, EPA rules require action to be taken when the percentage of lead in drinking water exceeds {0.0000015\%} (15 parts per billion). At the opposite extreme, recycling contamination rates as high as {10\%} are often considered acceptable.

Because of all the very different ways in which percentages could be used, I think it may make sense to propose an alternate system of units to measure one class of probabilities, namely the probabilities of avoiding some highly undesirable outcome, such as death, accident or illness. The units I propose are that of “nines“, which are already commonly used to measure availability of some service or purity of a material, but can be equally used to measure the safety (i.e., lack of risk) of some activity. Informally, nines measure how many consecutive appearances of the digit {9} are in the probability of successfully avoiding the negative outcome, thus

  • {90\%} success = one nine of safety
  • {99\%} success = two nines of safety
  • {99.9\%} success = three nines of safety
and so forth. Using the mathematical device of logarithms, one can also assign a fractional number of nines of safety to a general probability:

Definition 1 (Nines of safety) An activity (affecting one or more persons, over some given period of time) that has a probability {p} of the “safe” outcome and probability {1-p} of the “unsafe” outcome will have {k} nines of safety against the unsafe outcome, where {k} is defined by the formula

\displaystyle  k = -\log_{10}(1-p) \ \ \ \ \ (1)

(where {\log_{10}} is the logarithm to base ten), or equivalently

\displaystyle  p = 1 - 10^{-k}. \ \ \ \ \ (2)

Remark 2 Because of the various uncertainties in measuring probabilities, as well as the inaccuracies in some of the assumptions and approximations we will be making later, we will not attempt to measure the number of nines of safety beyond the first decimal point; thus we will round to the nearest tenth of a nine of safety throughout this post.

Here is a conversion table between percentage rates of success (the safe outcome), failure (the unsafe outcome), and the number of nines of safety one has:

Success rate {p} Failure rate {1-p} Number of nines {k}
{0\%} {100\%} {0.0}
{50\%} {50\%} {0.3}
{75\%} {25\%} {0.6}
{80\%} {20\%} {0.7}
{90\%} {10\%} {1.0}
{95\%} {5\%} {1.3}
{97.5\%} {2.5\%} {1.6}
{98\%} {2\%} {1.7}
{99\%} {1\%} {2.0}
{99.5\%} {0.5\%} {2.3}
{99.75\%} {0.25\%} {2.6}
{99.8\%} {0.2\%} {2.7}
{99.9\%} {0.1\%} {3.0}
{99.95\%} {0.05\%} {3.3}
{99.975\%} {0.025\%} {3.6}
{99.98\%} {0.02\%} {3.7}
{99.99\%} {0.01\%} {4.0}
{100\%} {0\%} infinite

Thus, if one has no nines of safety whatsoever, one is guaranteed to fail; but each nine of safety one has reduces the failure rate by a factor of {10}. In an ideal world, one would have infinitely many nines of safety against any risk, but in practice there are no {100\%} guarantees against failure, and so one can only expect a finite amount of nines of safety in any given situation. Realistically, one should thus aim to have as many nines of safety as one can reasonably expect to have, but not to demand an infinite amount.

Remark 3 The number of nines of safety against a certain risk is not absolute; it will depend not only on the risk itself, but (a) the number of people exposed to the risk, and (b) the length of time one is exposed to the risk. Exposing more people or increasing the duration of exposure will reduce the number of nines, and conversely exposing fewer people or reducing the duration will increase the number of nines; see Proposition 7 below for a rough rule of thumb in this regard.

Remark 4 Nines of safety are a logarithmic scale of measurement, rather than a linear scale. Other familiar examples of logarithmic scales of measurement include the Richter scale of earthquake magnitude, the pH scale of acidity, the decibel scale of sound level, octaves in music, and the magnitude scale for stars.

Remark 5 One way to think about nines of safety is via the Swiss cheese model that was created recently to describe pandemic risk management. In this model, each nine of safety can be thought of as a slice of Swiss cheese, with holes occupying {10\%} of that slice. Having {k} nines of safety is then analogous to standing behind {k} such slices of Swiss cheese. In order for a risk to actually impact you, it must pass through each of these {k} slices. A fractional nine of safety corresponds to a fractional slice of Swiss cheese that covers the amount of space given by the above table. For instance, {0.6} nines of safety corresponds to a fractional slice that covers about {75\%} of the given area (leaving {25\%} uncovered).

Now to give some real-world examples of nines of safety. Using data for deaths in the US in 2019 (without attempting to account for factors such as age and gender), a random US citizen will have had the following amount of safety from dying from some selected causes in that year:

Cause of death Mortality rate per {100,\! 000} (approx.) Nines of safety
All causes {870} {2.0}
Heart disease {200} {2.7}
Cancer {180} {2.7}
Accidents {52} {3.3}
Drug overdose {22} {3.7}
Influenza/Pneumonia {15} {3.8}
Suicide {14} {3.8}
Gun violence {12} {3.9}
Car accident {11} {4.0}
Murder {5} {4.3}
Airplane crash {0.14} {5.9}
Lightning strike {0.006} {7.2}

The safety of air travel is particularly remarkable: a given hour of flying in general aviation has a fatality rate of {0.00001}, or about {5} nines of safety, while for the major carriers the fatality rate drops down to {0.0000005}, or about {7.3} nines of safety.

Of course, in 2020, COVID-19 deaths became significant. In this year in the US, the mortality rate for COVID-19 (as the underlying or contributing cause of death) was {91.5} per {100,\! 000}, corresponding to {3.0} nines of safety, which was less safe than all other causes of death except for heart disease and cancer. At this time of writing, data for all of 2021 is of course not yet available, but it seems likely that the safety level would be even lower for this year.

Some further illustrations of the concept of nines of safety:

  • Each round of Russian roulette has a success rate of {5/6}, providing only {0.8} nines of safety. Of course, the safety will decrease with each additional round: one has only {0.5} nines of safety after two rounds, {0.4} nines after three rounds, and so forth. (See also Proposition 7 below.)
  • The ancient Roman punishment of decimation, by definition, provided exactly one nine of safety to each soldier being punished.
  • Rolling a {1} on a {20}-sided die is a risk that carries about {1.3} nines of safety.
  • Rolling a double one (“snake eyes“) from two six-sided dice carries about {1.6} nines of safety.
  • One has about {2.6} nines of safety against the risk of someone randomly guessing your birthday on the first attempt.
  • A null hypothesis has {1.3} nines of safety against producing a {p = 0.05} statistically significant result, and {2.0} nines against producing a {p=0.01} statistically significant result. (However, one has to be careful when reversing the conditional; a {p=0.01} statistically significant result does not necessarily have {2.0} nines of safety against the null hypothesis. In Bayesian statistics, the precise relationship between the two risks is given by Bayes’ theorem.)
  • If a poker opponent is dealt a five-card hand, one has {5.8} nines of safety against that opponent being dealt a royal flush, {4.8} against a straight flush or higher, {3.6} against four-of-a-kind or higher, {2.8} against a full house or higher, {2.4} against a flush or higher, {2.1} against a straight or higher, {1.5} against three-of-a-kind or higher, {1.1} against two pairs or higher, and just {0.3} against one pair or higher. (This data was converted from this Wikipedia table.)
  • A {k}-digit PIN number (or a {k}-digit combination lock) carries {k} nines of safety against each attempt to randomly guess the PIN. A length {k} password that allows for numbers, upper and lower case letters, and punctuation carries about {2k} nines of safety against a single guess. (For the reduction in safety caused by multiple guesses, see Proposition 7 below.)

Here is another way to think about nines of safety:

Proposition 6 (Nines of safety extend expected onset of risk) Suppose a certain risky activity has {k} nines of safety. If one repeatedly indulges in this activity until the risk occurs, then the expected number of trials before the risk occurs is {10^k}.

Proof: The probability that the risk is activated after exactly {n} trials is {(1-10^{-k})^{n-1} 10^{-k}}, which is a geometric distribution of parameter {10^{-k}}. The claim then follows from the standard properties of that distribution. \Box

Thus, for instance, if one performs some risky activity daily, then the expected length of time before the risk occurs is given by the following table:

Daily nines of safety Expected onset of risk
{0} One day
{0.8} One week
{1.5} One month
{2.6} One year
{2.9} Two years
{3.3} Five years
{3.6} Ten years
{3.9} Twenty years
{4.3} Fifty years
{4.6} A century

Or, if one wants to convert the yearly risks of dying from a specific cause into expected years before that cause of death would occur (assuming for sake of discussion that no other cause of death exists):

Yearly nines of safety Expected onset of risk
{0} One year
{0.3} Two years
{0.7} Five years
{1} Ten years
{1.3} Twenty years
{1.7} Fifty years
{2.0} A century

These tables suggest a relationship between the amount of safety one would have in a short timeframe, such as a day, and a longer time frame, such as a year. Here is an approximate formalisation of that relationship:

Proposition 7 (Repeated exposure reduces nines of safety) If a risky activity with {k} nines of safety is (independently) repeated {m} times, then (assuming {k} is large enough depending on {m}), the repeated activity will have approximately {k - \log_{10} m} nines of safety. Conversely: if the repeated activity has {k'} nines of safety, the individual activity will have approximately {k' + \log_{10} m} nines of safety.

Proof: An activity with {k} nines of safety will be safe with probability {1-10^{-k}}, hence safe with probability {(1-10^{-k})^m} if repeated independently {m} times. For {k} large, we can approximate

\displaystyle  (1 - 10^{-k})^m \approx 1 - m 10^{-k} = 1 - 10^{-(k - \log_{10} m)}

giving the former claim. The latter claim follows from inverting the former. \Box

Remark 8 The hypothesis of independence here is key. If there is a lot of correlation between the risks between different repetitions of the activity, then there can be much less reduction in safety caused by that repetition. As a simple example, suppose that {90\%} of a workforce are trained to perform some task flawlessly no matter how many times they repeat the task, but the remaining {10\%} are untrained and will always fail at that task. If one selects a random worker and asks them to perform the task, one has {1.0} nines of safety against the task failing. If one took that same random worker and asked them to perform the task {m} times, the above proposition might suggest that the number of nines of safety would drop to approximately {1.0 - \log_{10} m}; but in this case there is perfect correlation, and in fact the number of nines of safety remains steady at {1.0} since it is the same {10\%} of the workforce that would fail each time.

Because of this caveat, one should view the above proposition as only a crude first approximation that can be used as a simple rule of thumb, but should not be relied upon for more precise calculations.

One can repeat a risk either in time (extending the time of exposure to the risk, say from a day to a year), or in space (by exposing the risk to more people). The above proposition then gives an additive conversion law for nines of safety in either case. Here are some conversion tables for time:

From/to Daily Weekly Monthly Yearly
Daily 0 -0.8 -1.5 -2.6
Weekly +0.8 0 -0.6 -1.7
Monthly +1.5 +0.6 0 -1.1
Yearly +2.6 +1.7 +1.1 0

From/to Yearly Per 5 yr Per decade Per century
Yearly 0 -0.7 -1.0 -2.0
Per 5 yr +0.7 0 -0.3 -1.3
Per decade +1.0 + -0.3 0 -1.0
Per century +2.0 +1.3 +1.0 0

For instance, as mentioned before, the yearly amount of safety against cancer is about {2.7}. Using the above table (and making the somewhat unrealistic hypothesis of independence), we then predict the daily amount of safety against cancer to be about {2.7 + 2.6 = 5.3} nines, the weekly amount to be about {2.7 + 1.7 = 4.4} nines, and the amount of safety over five years to drop to about {2.7 - 0.7 = 2.0} nines.

Now we turn to conversions in space. If one knows the level of safety against a certain risk for an individual, and then one (independently) exposes a group of such individuals to that risk, then the reduction in nines of safety when considering the possibility that at least one group member experiences this risk is given by the following table:

Group Reduction in safety
You ({1} person) {0}
You and your partner ({2} people) {-0.3}
You and your parents ({3} people) {-0.5}
You, your partner, and three children ({5} people) {-0.7}
An extended family of {10} people {-1.0}
A class of {30} people {-1.5}
A workplace of {100} people {-2.0}
A school of {1,\! 000} people {-3.0}
A university of {10,\! 000} people {-4.0}
A town of {100,\! 000} people {-5.0}
A city of {1} million people {-6.0}
A state of {10} million people {-7.0}
A country of {100} million people {-8.0}
A continent of {1} billion people {-9.0}
The entire planet {-9.8}

For instance, in a given year (and making the somewhat implausible assumption of independence), you might have {2.7} nines of safety against cancer, but you and your partner collectively only have about {2.7 - 0.3 = 2.4} nines of safety against this risk, your family of five might only have about {2.7 - 0.7 = 2} nines of safety, and so forth. By the time one gets to a group of {1,\! 000} people, it actually becomes very likely that at least one member of the group will die of cancer in that year. (Here the precise conversion table breaks down, because a negative number of nines such as {2.7 - 3.0 = -0.3} is not possible, but one should interpret a prediction of a negative number of nines as an assertion that failure is very likely to happen. Also, in practice the reduction in safety is less than this rule predicts, due to correlations such as risk factors that are common to the group being considered that are incompatible with the assumption of independence.)

In the opposite direction, any reduction in exposure (either in time or space) to a risk will increase one’s safety level, as per the following table:

Reduction in exposure Additional nines of safety
{\div 1} {0}
{\div 2} {+0.3}
{\div 3} {+0.5}
{\div 5} {+0.7}
{\div 10} {+1.0}
{\div 100} {+2.0}

For instance, a five-fold reduction in exposure will reclaim about {0.7} additional nines of safety.

Here is a slightly different way to view nines of safety:

Proposition 9 Suppose that a group of {m} people are independently exposed to a given risk. If there are at most

\displaystyle  \log_{10} \frac{1}{1-2^{-1/m}}

nines of individual safety against that risk, then there is at least a {50\%} chance that one member of the group is affected by the risk.

Proof: If individually there are {k} nines of safety, then the probability that all the members of the group avoid the risk is {(1-10^{-k})^m}. Since the inequality

\displaystyle  (1-10^{-k})^m \leq \frac{1}{2}

is equivalent to

\displaystyle  k \leq \log_{10} \frac{1}{1-2^{-1/m}},

the claim follows. \Box

Thus, for a group to collectively avoid a risk with at least a {50\%} chance, one needs the following level of individual safety:

Group Individual safety level required
You ({1} person) {0.3}
You and your partner ({2} people) {0.5}
You and your parents ({3} people) {0.7}
You, your partner, and three children ({5} people) {0.9}
An extended family of {10} people {1.2}
A class of {30} people {1.6}
A workplace of {100} people {2.2}
A school of {1,\! 000} people {3.2}
A university of {10,\! 000} people {4.2}
A town of {100,\! 000} people {5.2}
A city of {1} million people {6.2}
A state of {10} million people {7.2}
A country of {100} million people {8.2}
A continent of {1} billion people {9.2}
The entire planet {10.0}

For large {m}, the level {k} of nines of individual safety required to protect a group of size {m} with probability at least {50\%} is approximately {\log_{10} \frac{m}{\ln 2} \approx (\log_{10} m) + 0.2}.

Precautions that can work to prevent a certain risk from occurring will add additional nines of safety against that risk, even if the precaution is not {100\%} effective. Here is the precise rule:

Proposition 10 (Precautions add nines of safety) Suppose an activity carries {k} nines of safety against a certain risk, and a separate precaution can independently protect against that risk with {l} nines of safety (that is to say, the probability that the protection is effective is {1 - 10^{-l}}). Then applying that precaution increases the number of nines in the activity from {k} to {k+l}.

Proof: The probability that the precaution fails and the risk then occurs is {10^{-l} \times 10^{-k} = 10^{-(k+l)}}. The claim now follows from Definition 1. \Box

In particular, we can repurpose the table at the start of this post as a conversion chart for effectiveness of a precaution:

Effectiveness Failure rate Additional nines provided
{0\%} {100\%} {+0.0}
{50\%} {50\%} {+0.3}
{75\%} {25\%} {+0.6}
{80\%} {20\%} {+0.7}
{90\%} {10\%} {+1.0}
{95\%} {5\%} {+1.3}
{97.5\%} {2.5\%} {+1.6}
{98\%} {2\%} {+1.7}
{99\%} {1\%} {+2.0}
{99.5\%} {0.5\%} {+2.3}
{99.75\%} {0.25\%} {+2.6}
{99.8\%} {0.2\%} {+2.7}
{99.9\%} {0.1\%} {+3.0}
{99.95\%} {0.05\%} {+3.3}
{99.975\%} {0.025\%} {+3.6}
{99.98\%} {0.02\%} {+3.7}
{99.99\%} {0.01\%} {+4.0}
{100\%} {0\%} infinite

Thus for instance a precaution that is {80\%} effective will add {0.7} nines of safety, a precaution that is {99.8\%} effective will add {2.7} nines of safety, and so forth. The mRNA COVID vaccines by Pfizer and Moderna have somewhere between {88\% - 96\%} effectiveness against symptomatic COVID illness, providing about {0.9-1.4} nines of safety against that risk, and over {95\%} effectiveness against severe illness, thus adding at least {1.3} nines of safety in this regard.

A slight variant of the above rule can be stated using the concept of relative risk:

Proposition 11 (Relative risk and nines of safety) Suppose an activity carries {k} nines of safety against a certain risk, and an action multiplies the chance of failure by some relative risk {R}. Then the action removes {\log_{10} R} nines of safety (if {R > 1}) or adds {-\log_{10} R} nines of safety (if {R<1}) to the original activity.

Proof: The additional action adjusts the probability of failure from {10^{-k}} to {R \times 10^{-k} = 10^{-(k - \log_{10} R)}}. The claim now follows from Definition 1. \Box

Here is a conversion chart between relative risk and change in nines of safety:

Relative risk Change in nines of safety
{0.01} {+2.0}
{0.02} {+1.7}
{0.05} {+1.3}
{0.1} {+1.0}
{0.2} {+0.7}
{0.5} {+0.3}
{1} {0}
{2} {-0.3}
{5} {-0.7}
{10} {-1.0}
{20} {-1.3}
{50} {-1.7}
{100} {-2.0}

Some examples:

  • Smoking increases the fatality rate of lung cancer by a factor of about {20}, thus removing about {1.3} nines of safety from this particular risk; it also increases the fatality rates of several other diseases, though not quite as dramatically an extent.
  • Seatbelts reduce the fatality rate in car accidents by a factor of about two, adding about {0.3} nines of safety. Airbags achieve a reduction of about {30-50\%}, adding about {0.2-0.3} additional nines of safety.
  • As far as transmission of COVID is concerned, it seems that constant use of face masks reduces transmission by a factor of about five (thus adding about {0.7} nines of safety), and similarly for constant adherence to social distancing; whereas for instance a {30\%} compliance with mask usage reduced transmission by about {10\%} (adding only {0.05} or so nines of safety).

The effect of combining multiple (independent) precautions together is cumulative; one can achieve quite a high level of safety by stacking together several precautions that individually have relatively low levels of effectiveness. Again, see the “swiss cheese model” referred to in Remark 5. For instance, if face masks add {0.7} nines of safety against contracting COVID, social distancing adds another {0.7} nines, and the vaccine provide another {1.0} nine of safety, implementing all three mitigation methods would (assuming independence) add a net of {2.4} nines of safety against contracting COVID.

In summary, when debating the value of a given risk mitigation measure, the correct question to ask is not quite “Is it certain to work” or “Can it fail?”, but rather “How many extra nines of safety does it add?”.

As one final comparison between nines of safety and other standard risk measures, we give the following proposition regarding large deviations from the mean.

Proposition 12 Let {X} be a normally distributed random variable of standard deviation {\sigma}, and let {\lambda > 0}. Then the “one-sided risk” of {X} exceeding its mean {{\bf E} X} by at least {\lambda \sigma} (i.e., {X \geq {\bf E} X + \lambda \sigma}) carries

\displaystyle  -\log_{10} \frac{1 - \mathrm{erf}(\lambda/\sqrt{2})}{2}

nines of safety, the “two-sided risk” of {X} deviating (in either direction) from its mean by at least {\lambda \sigma} (i.e., {|X-{\bf E} X| \geq \lambda \sigma}) carries

\displaystyle  -\log_{10} (1 - \mathrm{erf}(\lambda/\sqrt{2}))

nines of safety, where {\mathrm{erf}} is the error function.

Proof: This is a routine calculation using the cumulative distribution function of the normal distribution. \Box

Here is a short table illustrating this proposition:

Number {\lambda} of deviations from the mean One-sided nines of safety Two-sided nines of safety
{0} {0.3} {0.0}
{1} {0.8} {0.5}
{2} {1.6} {1.3}
{3} {2.9} {2.6}
{4} {4.5} {4.2}
{5} {6.5} {6.2}
{6} {9.0} {8.7}

Thus, for instance, the risk of a five sigma event (deviating by more than five standard deviations from the mean in either direction) should carry {6.2} nines of safety assuming a normal distribution, and so one would ordinarily feel extremely safe against the possibility of such an event, unless one started doing hundreds of thousands of trials. (However, we caution that this conclusion relies heavily on the assumption that one has a normal distribution!)

See also this older essay I wrote on anonymity on the internet, using bits as a measure of anonymity in much the same way that nines are used here as a measure of safety.

Joni Teräväinen and I have just uploaded to the arXiv our preprint “The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero“. This paper is a development of the theme that certain conjectures in analytic number theory become easier if one makes the hypothesis that Siegel zeroes exist; this places one in a presumably “illusory” universe, since the widely believed Generalised Riemann Hypothesis (GRH) precludes the existence of such zeroes, yet this illusory universe seems remarkably self-consistent and notoriously impossible to eliminate from one’s analysis.

For the purposes of this paper, a Siegel zero is a zero {\beta} of a Dirichlet {L}-function {L(\cdot,\chi)} corresponding to a primitive quadratic character {\chi} of some conductor {q_\chi}, which is close to {1} in the sense that

\displaystyle  \beta = 1 - \frac{1}{\eta \log q_\chi}

for some large {\eta \gg 1} (which we will call the quality) of the Siegel zero. The significance of these zeroes is that they force the Möbius function {\mu} and the Liouville function {\lambda} to “pretend” to be like the exceptional character {\chi} for primes of magnitude comparable to {q_\chi}. Indeed, if we define an exceptional prime to be a prime {p^*} in which {\chi(p^*) \neq -1}, then very few primes near {q_\chi} will be exceptional; in our paper we use some elementary arguments to establish the bounds

\displaystyle  \sum_{q_\chi^{1/2+\varepsilon} < p^* \leq x} \frac{1}{p^*} \ll_\varepsilon \frac{\log x}{\eta \log q_\chi} \ \ \ \ \ (1)

for any {x \geq q_\chi^{1/2+\varepsilon}} and {\varepsilon>0}, where the sum is over exceptional primes in the indicated range {q_\chi^{1/2+\varepsilon} < p^* \leq x}; this bound is non-trivial for {x} as large as {q_\chi^{\eta^{1-\varepsilon}}}. (See Section 1 of this blog post for some variants of this argument, which were inspired by work of Heath-Brown.) There is also a companion bound (somewhat weaker) that covers a range of {p^*} a little bit below {q_\chi^{1/2}}.

One of the early influential results in this area was the following result of Heath-Brown, which I previously blogged about here:

Theorem 1 (Hardy-Littlewood assuming Siegel zero) Let {h} be a fixed natural number. Suppose one has a Siegel zero {\beta} associated to some conductor {q_\chi}. Then we have

\displaystyle  \sum_{n \leq x} \Lambda(n) \Lambda(n+h) = ({\mathfrak S} + O( \frac{1}{\log\log \eta} )) x

for all {q_\chi^{250} \leq x \leq q_\chi^{300}}, where {\Lambda} is the von Mangoldt function and {{\mathfrak S}} is the singular series

\displaystyle  {\mathfrak S} = \prod_{p|h} \frac{p}{p-1} \prod_{p \nmid h} (1 - \frac{1}{(p-1)^2})

In particular, Heath-Brown showed that if there are infinitely many Siegel zeroes, then there are also infinitely many twin primes, with the correct asymptotic predicted by the Hardy-Littlewood prime tuple conjecture at infinitely many scales.

Very recently, Chinis established an analogous result for the Chowla conjecture (building upon earlier work of Germán and Katai):

Theorem 2 (Chowla assuming Siegel zero) Let {h_1,\dots,h_\ell} be distinct fixed natural numbers. Suppose one has a Siegel zero {\beta} associated to some conductor {q_\chi}. Then one has

\displaystyle  \sum_{n \leq x} \lambda(n+h_1) \dots \lambda(n+h_\ell) \ll \frac{x}{(\log\log \eta)^{1/2} (\log \eta)^{1/12}}

in the range {q_\chi^{10} \leq x \leq q_\chi^{\log\log \eta / 3}}, where {\lambda} is the Liouville function.

In our paper we unify these results and also improve the quantitative estimates and range of {x}:

Theorem 3 (Hardy-Littlewood-Chowla assuming Siegel zero) Let {h_1,\dots,h_k,h'_1,\dots,h'_\ell} be distinct fixed natural numbers with {k \leq 2}. Suppose one has a Siegel zero {\beta} associated to some conductor {q_\chi}. Then one has

\displaystyle  \sum_{n \leq x} \Lambda(n+h_1) \dots \Lambda(n+h_k) \lambda(n+h'_1) \dots \lambda(n+h'_\ell)

\displaystyle = ({\mathfrak S} + O_\varepsilon( \frac{1}{\log^{1/10\max(1,k)} \eta} )) x

for

\displaystyle  q_\chi^{10k+\frac{1}{2}+\varepsilon} \leq x \leq q_\chi^{\eta^{1/2}}

for any fixed {\varepsilon>0}.

Our argument proceeds by a series of steps in which we replace {\Lambda} and {\lambda} by more complicated looking, but also more tractable, approximations, until the correlation is one that can be computed in a tedious but straightforward fashion by known techniques. More precisely, the steps are as follows:

  • (i) Replace the Liouville function {\lambda} with an approximant {\lambda_{\mathrm{Siegel}}}, which is a completely multiplicative function that agrees with {\lambda} at small primes and agrees with {\chi} at large primes.
  • (ii) Replace the von Mangoldt function {\Lambda} with an approximant {\Lambda_{\mathrm{Siegel}}}, which is the Dirichlet convolution {\chi * \log} multiplied by a Selberg sieve weight {\nu} to essentially restrict that convolution to almost primes.
  • (iii) Replace {\lambda_{\mathrm{Siegel}}} with a more complicated truncation {\lambda_{\mathrm{Siegel}}^\sharp} which has the structure of a “Type I sum”, and which agrees with {\lambda_{\mathrm{Siegel}}} on numbers that have a “typical” factorization.
  • (iv) Replace the approximant {\Lambda_{\mathrm{Siegel}}} with a more complicated approximant {\Lambda_{\mathrm{Siegel}}^\sharp} which has the structure of a “Type I sum”.
  • (v) Now that all terms in the correlation have been replaced with tractable Type I sums, use standard Euler product calculations and Fourier analysis, similar in spirit to the proof of the pseudorandomness of the Selberg sieve majorant for the primes in this paper of Ben Green and myself, to evaluate the correlation to high accuracy.

Steps (i), (ii) proceed mainly through estimates such as (1) and standard sieve theory bounds. Step (iii) is based primarily on estimates on the number of smooth numbers of a certain size.

The restriction {k \leq 2} in our main theorem is needed only to execute step (iv) of this step. Roughly speaking, the Siegel approximant {\Lambda_{\mathrm{Siegel}}} to {\Lambda} is a twisted, sieved version of the divisor function {\tau}, and the types of correlation one is faced with at the start of step (iv) are a more complicated version of the divisor correlation sum

\displaystyle  \sum_{n \leq x} \tau(n+h_1) \dots \tau(n+h_k).

For {k=1} this sum can be easily controlled by the Dirichlet hyperbola method. For {k=2} one needs the fact that {\tau} has a level of distribution greater than {1/2}; in fact Kloosterman sum bounds give a level of distribution of {2/3}, a folklore fact that seems to have first been observed by Linnik and Selberg. We use a (notationally more complicated) version of this argument to treat the sums arising in (iv) for {k \leq 2}. Unfortunately for {k > 2} there are no known techniques to unconditionally obtain asymptotics, even for the model sum

\displaystyle  \sum_{n \leq x} \tau(n) \tau(n+1) \tau(n+2),

although we do at least have fairly convincing conjectures as to what the asymptotics should be. Because of this, it seems unlikely that one will be able to relax the {k \leq 2} hypothesis in our main theorem at our current level of understanding of analytic number theory.

Step (v) is a tedious but straightforward sieve theoretic computation, similar in many ways to the correlation estimates of Goldston and Yildirim used in their work on small gaps between primes (as discussed for instance here), and then also used by Ben Green and myself to locate arithmetic progressions in primes.

A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do not seem to be trivially resolvable.

  1. Does there exist a smooth function {f: {\bf R} \rightarrow {\bf R}} which is nowhere analytic, but is such that the Taylor series {\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n} converges for every {x, x_0 \in {\bf R}}? (Of course, this series would not then converge to {f}, but instead to some analytic function {f_{x_0}(x)} for each {x_0}.) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort. (Update: answered by Alexander Shaposhnikov in comments.)
  2. Is there a function {f: {\bf R} \rightarrow {\bf R}} which meets every polynomial {P: {\bf R} \rightarrow {\bf R}} to infinite order in the following sense: for every polynomial {P}, there exists {x_0} such that {f^{(n)}(x_0) = P^{(n)}(x_0)} for all {n=0,1,2,\dots}? Such a function would be rather pathological, perhaps resembling a space-filling curve. (Update: solved for smooth {f} by Aleksei Kulikov in comments. The situation currently remains unclear in the general case.)
  3. Is there a power series {\sum_{n=0}^\infty a_n x^n} that diverges everywhere (except at {x=0}), but which becomes pointwise convergent after dividing each of the monomials {a_n x^n} into pieces {a_n x^n = \sum_{j=1}^\infty a_{n,j} x^n} for some {a_{n,j}} summing absolutely to {a_n}, and then rearranging, i.e., there is some rearrangement {\sum_{m=1}^\infty a_{n_m, j_m} x^{n_m}} of {\sum_{n=0}^\infty \sum_{j=1}^\infty a_{n,j} x^n} that is pointwise convergent for every {x}? (Update: solved by Jacob Manaker in comments.)

Feel free to post answers or other thoughts on these questions in the comments.

Rachel Greenfeld and I have just uploaded to the arXiv our preprint “Undecidable translational tilings with only two tiles, or one nonabelian tile“. This paper studies the following question: given a finitely generated group {G}, a (periodic) subset {E} of {G}, and finite sets {F_1,\dots,F_J} in {G}, is it possible to tile {E} by translations {a_j+F_j} of the tiles {F_1,\dots,F_J}? That is to say, is there a solution {\mathrm{X}_1 = A_1, \dots, \mathrm{X}_J = A_J} to the (translational) tiling equation

\displaystyle  (\mathrm{X}_1 \oplus F_1) \uplus \dots \uplus (\mathrm{X}_J \oplus F_J) = E \ \ \ \ \ (1)

for some subsets {A_1,\dots,A_J} of {G}, where {A \oplus F} denotes the set of sums {\{a+f: a \in A, f \in F \}} if the sums {a+f} are all disjoint (and is undefined otherwise), and {\uplus} denotes disjoint union. (One can also write the tiling equation in the language of convolutions as {1_{\mathrm{X}_1} * 1_{F_1} + \dots + 1_{\mathrm{X}_J} * 1_{F_J} = 1_E}.)

A bit more specifically, the paper studies the decidability of the above question. There are two slightly different types of decidability one could consider here:

  • Logical decidability. For a given {G, E, J, F_1,\dots,F_J}, one can ask whether the solvability of the tiling equation (1) is provable or disprovable in ZFC (where we encode all the data {G, E, F_1,\dots,F_J} by appropriate constructions in ZFC). If this is the case we say that the tiling equation (1) (or more precisely, the solvability of this equation) is logically decidable, otherwise it is logically undecidable.
  • Algorithmic decidability. For data {G,E,J, F_1,\dots,F_J} in some specified class (and encoded somehow as binary strings), one can ask whether the solvability of the tiling equation (1) can be correctly determined for all choices of data in this class by the output of some Turing machine that takes the data as input (encoded as a binary string) and halts in finite time, returning either YES if the equation can be solved or NO otherwise. If this is the case, we say the tiling problem of solving (1) for data in the given class is algorithmically decidable, otherwise it is algorithmically undecidable.

Note that the notion of logical decidability is “pointwise” in the sense that it pertains to a single choice of data {G,E,J,F_1,\dots,F_J}, whereas the notion of algorithmic decidability pertains instead to classes of data, and is only interesting when this class is infinite. Indeed, any tiling problem with a finite class of data is trivially decidable because one could simply code a Turing machine that is basically a lookup table that returns the correct answer for each choice of data in the class. (This is akin to how a student with a good memory could pass any exam if the questions are drawn from a finite list, merely by memorising an answer key for that list of questions.)

The two notions are related as follows: if a tiling problem (1) is algorithmically undecidable for some class of data, then the tiling equation must be logically undecidable for at least one choice of data for this class. For if this is not the case, one could algorithmically decide the tiling problem by searching for proofs or disproofs that the equation (1) is solvable for a given choice of data; the logical decidability of all such solvability questions will ensure that this algorithm always terminates in finite time.

One can use the Gödel completeness theorem to interpret logical decidability in terms of universes (also known as structures or models) of ZFC. In addition to the “standard” universe {{\mathfrak U}} of sets that we believe satisfies the axioms of ZFC, there are also other “nonstandard” universes {{\mathfrak U}^*} that also obey the axioms of ZFC. If the solvability of a tiling equation (1) is logically undecidable, this means that such a tiling exists in some universes of ZFC, but not in others.

(To continue the exam analogy, we thus see that a yes-no exam question is logically undecidable if the answer to the question is yes in some parallel universes, but not in others. A course syllabus is algorithmically undecidable if there is no way to prepare for the final exam for the course in a way that guarantees a perfect score (in the standard universe).)

Questions of decidability are also related to the notion of aperiodicity. For a given {G, E, J, F_1,\dots,F_J}, a tiling equation (1) is said to be aperiodic if the equation (1) is solvable (in the standard universe {{\mathfrak U}} of ZFC), but none of the solutions (in that universe) are completely periodic (i.e., there are no solutions {\mathrm{X}_1 = A_1,\dots, \mathrm{X}_J = A_J} where all of the {A_1,\dots,A_J} are periodic). Perhaps the most well-known example of an aperiodic tiling (in the context of {{\bf R}^2}, and using rotations as well as translations) come from the Penrose tilings, but there are many others besides.

It was (essentially) observed by Hao Wang in the 1960s that if a tiling equation is logically undecidable, then it must necessarily be aperiodic. Indeed, if a tiling equation fails to be aperiodic, then (in the standard universe) either there is a periodic tiling, or there are no tilings whatsoever. In the former case, the periodic tiling can be used to give a finite proof that the tiling equation is solvable; in the latter case, the compactness theorem implies that there is some finite fragment of {E} that is not compatible with being tiled by {F_1,\dots,F_J}, and this provides a finite proof that the tiling equation is unsolvable. Thus in either case the tiling equation is logically decidable.

This observation of Wang clarifies somewhat how logically undecidable tiling equations behave in the various universes of ZFC. In the standard universe, tilings exist, but none of them will be periodic. In nonstandard universes, tilings may or may not exist, and the tilings that do exist may be periodic (albeit with a nonstandard period); but there must be at least one universe in which no tiling exists at all.

In one dimension when {G={\bf Z}} (or more generally {G = {\bf Z} \times G_0} with {G_0} a finite group), a simple pigeonholing argument shows that no tiling equations are aperiodic, and hence all tiling equations are decidable. However the situation changes in two dimensions. In 1966, Berger (a student of Wang) famously showed that there exist tiling equations (1) in the discrete plane {E = G = {\bf Z}^2} that are aperiodic, or even logically undecidable; in fact he showed that the tiling problem in this case (with arbitrary choices of data {J, F_1,\dots,F_J}) was algorithmically undecidable. (Strictly speaking, Berger established this for a variant of the tiling problem known as the domino problem, but later work of Golomb showed that the domino problem could be easily encoded within the tiling problem.) This was accomplished by encoding the halting problem for Turing machines into the tiling problem (or domino problem); the latter is well known to be algorithmically undecidable (and thus have logically undecidable instances), and so the latter does also. However, the number of tiles {J} required for Berger’s construction was quite large: his construction of an aperiodic tiling required {J = 20426} tiles, and his construction of a logically undecidable tiling required an even larger (and not explicitly specified) collection of tiles. Subsequent work by many authors did reduce the number of tiles required; in the {E=G={\bf Z}^2} setting, the current world record for the fewest number of tiles in an aperiodic tiling is {J=8} (due to Amman, Grunbaum, and Shephard) and for a logically undecidable tiling is {J=11} (due to Ollinger). On the other hand, it is conjectured (see Grunbaum-Shephard and Lagarias-Wang) that one cannot lower {J} all the way to {1}:

Conjecture 1 (Periodic tiling conjecture) If {E} is a periodic subset of a finitely generated abelian group {G}, and {F} is a finite subset of {G}, then the tiling equation {\mathrm{X} \oplus F = E} is not aperiodic.

This conjecture is known to be true in two dimensions (by work of Bhattacharya when {G=E={\bf Z}^2}, and more recently by us when {E \subset G = {\bf Z}^2}), but remains open in higher dimensions. By the preceding discussion, the conjecture implies that every tiling equation with a single tile is logically decidable, and the problem of whether a given periodic set can be tiled by a single tile is algorithmically decidable.

In this paper we show on the other hand that aperiodic and undecidable tilings exist when {J=2}, at least if one is permitted to enlarge the group {G} a bit:

Theorem 2 (Logically undecidable tilings)
  • (i) There exists a group {G} of the form {G = {\bf Z}^2 \times G_0} for some finite abelian {G_0}, a subset {E_0} of {G_0}, and finite sets {F_1, F_2 \subset G} such that the tiling equation {(\mathbf{X}_1 \oplus F_1) \uplus (\mathbf{X}_2 \oplus F_2) = {\bf Z}^2 \times E_0} is logically undecidable (and hence also aperiodic).
  • (ii) There exists a dimension {d}, a periodic subset {E} of {{\bf Z}^d}, and finite sets {F_1, F_2 \subset G} such that tiling equation {(\mathbf{X}_1 \oplus F_1) \uplus (\mathbf{X}_2 \oplus F_2) = E} is logically undecidable (and hence also aperiodic).
  • (iii) There exists a non-abelian finite group {G_0} (with the group law still written additively), a subset {E_0} of {G_0}, and a finite set {F \subset {\bf Z}^2 \times G_0} such that the nonabelian tiling equation {\mathbf{X} \oplus F = {\bf Z}^2 \times E_0} is logically undecidable (and hence also aperiodic).

We also have algorithmic versions of this theorem. For instance, the algorithmic version of (i) is that the problem of determining solvability of the tiling equation {(\mathbf{X}_1 \oplus F_1) \uplus (\mathbf{X}_2 \oplus F_2) = {\bf Z}^2 \times E_0} for a given choice of finite abelian group {G_0}, subset {E_0} of {G_0}, and finite sets {F_1, F_2 \subset {\bf Z}^2 \times G_0} is algorithmically undecidable. Similarly for (ii), (iii).

This result (together with a negative result discussed below) suggest to us that there is a significant qualitative difference in the {J=1} theory of tiling by a single (abelian) tile, and the {J \geq 2} theory of tiling with multiple tiles (or one non-abelian tile). (The positive results on the periodic tiling conjecture certainly rely heavily on the fact that there is only one tile, in particular there is a “dilation lemma” that is only available in this setting that is of key importance in the two dimensional theory.) It would be nice to eliminate the group {G_0} from (i) (or to set {d=2} in (ii)), but I think this would require a fairly significant modification of our methods.

Like many other undecidability results, the proof of Theorem 2 proceeds by a sequence of reductions, in which the undecidability of one problem is shown to follow from the undecidability of another, more “expressive” problem that can be encoded inside the original problem, until one reaches a problem that is so expressive that it encodes a problem already known to be undecidable. Indeed, all three undecidability results are ultimately obtained from Berger’s undecidability result on the domino problem.

The first step in increasing expressiveness is to observe that the undecidability of a single tiling equation follows from the undecidability of a system of tiling equations. More precisely, suppose we have non-empty finite subsets {F_j^{(m)}} of a finitely generated group {G} for {j=1,\dots,J} and {m=1,\dots,M}, as well as periodic sets {E^{(m)}} of {G} for {m=1,\dots,M}, such that it is logically undecidable whether the system of tiling equations

\displaystyle  (\mathrm{X}_1 \oplus F_1^{(m)}) \uplus \dots \uplus (\mathrm{X}_J \oplus F_J^{(m)}) = E^{(m)} \ \ \ \ \ (2)

for {m=1,\dots,M} has no solution {\mathrm{X}_1 = A_1,\dots, \mathrm{X}_J = A_J} in {G}. Then, for any {N>M}, we can “stack” these equations into a single tiling equation in the larger group {G \times {\bf Z}/N{\bf Z}}, and specifically to the equation

\displaystyle  (\mathrm{X}_1 \oplus F_1) \uplus \dots \uplus (\mathrm{X}_J \oplus F_J) = E \ \ \ \ \ (3)

where

\displaystyle  F_j := \biguplus_{m=1}^M F_j^{(m)} \times \{m\}

and

\displaystyle  E := \biguplus_{m=1}^M E^{(m)} \times \{m\}.

It is a routine exercise to check that the system of equations (2) admits a solution in {G} if and only if the single equation (3) admits a equation in {G \times {\bf Z}/N{\bf Z}}. Thus, to prove the undecidability of a single equation of the form (3) it suffices to establish undecidability of a system of the form (2); note here how the freedom to select the auxiliary group {G_0} is important here.

We view systems of the form (2) as belonging to a kind of “language” in which each equation in the system is a “sentence” in the language imposing additional constraints on a tiling. One can now pick and choose various sentences in this language to try to encode various interesting problems. For instance, one can encode the concept of a function {f: {\bf Z}^2 \rightarrow G_0} taking values in a finite group {G_0} as a single tiling equation

\displaystyle  \mathrm{X} \oplus (\{0\} \times G_0) = {\bf Z}^2 \times G_0 \ \ \ \ \ (4)

since the solutions to this equation are precisely the graphs

\displaystyle  \mathrm{X} = \{ (n, f(n)): n \in {\bf Z}^2 \}

of a function {f: {\bf Z}^2 \rightarrow G_0}. By adding more tiling equations to this equation to form a larger system, we can start imposing additional constraints on this function {f}. For instance, if {x+H} is a coset of some subgroup {H} of {G_0}, we can impose the additional equation

\displaystyle  \mathrm{X} \oplus (\{0\} \times H) = {\bf Z}^2 \times (x+H) \ \ \ \ \ (5)

to impose the additional constraint that {f(n) \in x+H} for all {n \in {\bf Z}^2}, if we desire. If {G_0} happens to contain two distinct elements {1, -1}, and {h \in {\bf Z}^2}, then the additional equation

\displaystyle  \mathrm{X} \oplus (\{0,h\} \times \{0\}) = {\bf Z}^2 \times \{-1,1\} \ \ \ \ \ (6)

imposes the additional constraints that {f(n) \in \{-1,1\}} for all {n \in {\bf Z}^2}, and additionally that

\displaystyle  f(n+h) = -f(n)

for all {n \in {\bf Z}^2}.

This begins to resemble the equations that come up in the domino problem. Here one has a finite set of Wang tiles – unit squares {T} where each of the four sides is colored with a color {c_N(T), c_S(T), c_E(T), c_W(T)} (corresponding to the four cardinal directions North, South, East, and West) from some finite set {{\mathcal C}} of colors. The domino problem is then to tile the plane with copies of these tiles in such a way that adjacent sides match. In terms of equations, one is seeking to find functions {c_N, c_S, c_E, c_W: {\bf Z}^2 \rightarrow {\mathcal C}} obeying the pointwise constraint

\displaystyle  (c_N(n), c_S(n), c_E(n), c_W(n)) \in {\mathcal W} \ \ \ \ \ (7)

for all {n \in {\bf Z}^2} where {{\mathcal W}} is the set of colors associated to the set of Wang tiles being used, and the matching constraints

\displaystyle  c_S(n+(0,1)) = c_N(n); \quad c_W(n+(1,0)) = c_E(n) \ \ \ \ \ (8)

for all {{\bf Z}^2}. As it turns out, the pointwise constraint (7) can be encoded by tiling equations that are fancier versions of (4), (5), (6) that involve only one unknown tiling set {{\mathrm X}}, but in order to encode the matching constraints (8) we were forced to introduce a second tile (or work with nonabelian tiling equations). This appears to be an inherent feature of the method, since we found a partial rigidity result for tilings of one tile in one dimension that obstructs this encoding strategy from working when one only has one tile available. The result is as follows:

Proposition 3 (Swapping property) Consider the solutions to a tiling equation

\displaystyle  \mathrm{X} \oplus F = E \ \ \ \ \ (9)

in a one-dimensional group {G = {\bf Z} \times G_0} (with {G_0} a finite abelian group, {F} finite, and {E} periodic). Suppose there are two solutions {\mathrm{X} = A_0, \mathrm{X} = A_1} to this equation that agree on the left in the sense that

\displaystyle A_0 \cap (\{0, -1, -2, \dots\} \times G_0) = A_1 \cap (\{0, -1, -2, \dots\} \times G_0).

For any function {\omega: {\bf Z} \rightarrow \{0,1\}}, define the “swap” {A_\omega} of {A_0} and {A_1} to be the set

\displaystyle  A_\omega := \{ (n, g): n \in {\bf Z}, (n,g) \in A_{\omega(n)} \}

Then {A_\omega} also solves the equation (9).

One can think of {A_0} and {A_1} as “genes” with “nucleotides” {\{ g \in G_0: (n,g) \in A_0\}}, {\{ g \in G_0: (n,g) \in A_1\}} at each position {n \in {\bf Z}}, and {A_\omega} is a new gene formed by choosing one of the nucleotides from the “parent” genes {A_0}, {A_1} at each position. The above proposition then says that the solutions to the equation (9) must be closed under “genetic transfer” among any pair of genes that agree on the left. This seems to present an obstruction to trying to encode equation such as

\displaystyle  c(n+1) = c'(n)

for two functions {c, c': {\bf Z} \rightarrow \{-1,1\}} (say), which is a toy version of the matching constraint (8), since the class of solutions to this equation turns out not to obey this swapping property. On the other hand, it is easy to encode such equations using two tiles instead of one, and an elaboration of this construction is used to prove our main theorem.

Louis Esser, Burt Totaro, Chengxi Wang, and myself have just uploaded to the arXiv our preprint “Varieties of general type with many vanishing plurigenera, and optimal sine and sawtooth inequalities“. This is an interdisciplinary paper that arose because in order to optimize a certain algebraic geometry construction it became necessary to solve a purely analytic question which, while simple, did not seem to have been previously studied in the literature. We were able to solve the analytic question exactly and thus fully optimize the algebraic geometry construction, though the analytic question may have some independent interest.

Let us first discuss the algebraic geometry application. Given a smooth complex {n}-dimensional projective variety {X} there is a standard line bundle {K_X} attached to it, known as the canonical line bundle; {n}-forms on the variety become sections of this bundle. The bundle may not actually admit global sections; that is to say, the dimension {h^0(X, K_X)} of global sections may vanish. But as one raises the canonical line bundle {K_X} to higher and higher powers to form further line bundles {mK_X}, the number of global sections tends to increase; in particular, the dimension {h^0(X, mK_X)} of global sections (known as the {m^{th}} plurigenus) always obeys an asymptotic of the form

\displaystyle  h^0(X, mK_X) = \mathrm{vol}(X) \frac{m^n}{n!} + O( m^{n-1} )

as {m \rightarrow \infty} for some non-negative number {\mathrm{vol}(X)}, which is called the volume of the variety {X}, which is an invariant that reveals some information about the birational geometry of {X}. For instance, if the canonical line bundle is ample (or more generally, nef), this volume is equal to the intersection number {K_X^n} (roughly speaking, the number of common zeroes of {n} generic sections of the canonical line bundle); this is a special case of the asymptotic Riemann-Roch theorem. In particular, the volume {\mathrm{vol}(X)} is a natural number in this case. However, it is possible for the volume to also be fractional in nature. One can then ask: how small can the volume get {\mathrm{vol}(X)} without vanishing entirely? (By definition, varieties with non-vanishing volume are known as varieties of general type.)

It follows from a deep result obtained independently by Hacon–McKernan, Takayama and Tsuji that there is a uniform lower bound for the volume {\mathrm{vol}(X)} of all {n}-dimensional projective varieties of general type. However, the precise lower bound is not known, and the current paper is a contribution towards probing this bound by constructing varieties of particularly small volume in the high-dimensional limit {n \rightarrow \infty}. Prior to this paper, the best such constructions of {n}-dimensional varieties basically had exponentially small volume, with a construction of volume at most {e^{-(1+o(1))n \log n}} given by Ballico–Pignatelli–Tasin, and an improved construction with a volume bound of {e^{-\frac{1}{3} n \log^2 n}} given by Totaro and Wang. In this paper, we obtain a variant construction with the somewhat smaller volume bound of {e^{-(1-o(1)) n^{3/2} \log^{1/2} n}}; the method also gives comparable bounds for some other related algebraic geometry statistics, such as the largest {m} for which the pluricanonical map associated to the linear system {|mK_X|} is not a birational embedding into projective space.

The space {X} is constructed by taking a general hypersurface of a certain degree {d} in a weighted projective space {P(a_0,\dots,a_{n+1})} and resolving the singularities. These varieties are relatively tractable to work with, as one can use standard algebraic geometry tools (such as the ReidTai inequality) to provide sufficient conditions to guarantee that the hypersurface has only canonical singularities and that the canonical bundle is a reflexive sheaf, which allows one to calculate the volume exactly in terms of the degree {d} and weights {a_0,\dots,a_{n+1}}. The problem then reduces to optimizing the resulting volume given the constraints needed for the above-mentioned sufficient conditions to hold. After working with a particular choice of weights (which consist of products of mostly consecutive primes, with each product occuring with suitable multiplicities {c_0,\dots,c_{b-1}}), the problem eventually boils down to trying to minimize the total multiplicity {\sum_{j=0}^{b-1} c_j}, subject to certain congruence conditions and other bounds on the {c_j}. Using crude bounds on the {c_j} eventually leads to a construction with volume at most {e^{-0.8 n^{3/2} \log^{1/2} n}}, but by taking advantage of the ability to “dilate” the congruence conditions and optimizing over all dilations, we are able to improve the {0.8} constant to {1-o(1)}.

Now it is time to turn to the analytic side of the paper by describing the optimization problem that we solve. We consider the sawtooth function {g: {\bf R} \rightarrow (-1/2,1/2]}, with {g(x)} defined as the unique real number in {(-1/2,1/2]} that is equal to {x} mod {1}. We consider a (Borel) probability measure {\mu} on the real line, and then compute the average value of this sawtooth function

\displaystyle  \mathop{\bf E}_\mu g(x) := \int_{\bf R} g(x)\ d\mu(x)

as well as various dilates

\displaystyle  \mathop{\bf E}_\mu g(kx) := \int_{\bf R} g(kx)\ d\mu(x)

of this expectation. Since {g} is bounded above by {1/2}, we certainly have the trivial bound

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \frac{1}{2}.

However, this bound is not very sharp. For instance, the only way in which {\mathop{\bf E}_\mu g(x)} could attain the value of {1/2} is if the probability measure {\mu} was supported on half-integers, but in that case {\mathop{\bf E}_\mu g(2x)} would vanish. For the algebraic geometry application discussed above one is then led to the following question: for a given choice of {m}, what is the best upper bound {c^{\mathrm{saw}}_m} on the quantity {\min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx)} that holds for all probability measures {\mu}?

If one considers the deterministic case in which {\mu} is a Dirac mass supported at some real number {x_0}, then the Dirichlet approximation theorem tells us that there is {1 \leq k \leq m} such that {x_0} is within {\frac{1}{m+1}} of an integer, so we have

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \frac{1}{m+1}

in this case, and this bound is sharp for deterministic measures {\mu}. Thus we have

\displaystyle  \frac{1}{m+1} \leq c^{\mathrm{saw}}_m \leq \frac{1}{2}.

However, both of these bounds turn out to be far from the truth, and the optimal value of {c^{\mathrm{saw}}_m} is comparable to {\frac{\log 2}{\log m}}. In fact we were able to compute this quantity precisely:

Theorem 1 (Optimal bound for sawtooth inequality) Let {m \geq 1}.
  • (i) If {m = 2^r} for some natural number {r}, then {c^{\mathrm{saw}}_m = \frac{1}{r+2}}.
  • (ii) If {2^r < m \leq 2^{r+1}} for some natural number {r}, then {c^{\mathrm{saw}}_m = \frac{2^r}{2^r(r+1) + m}}.
In particular, we have {c^{\mathrm{saw}}_m = \frac{\log 2 + o(1)}{\log m}} as {m \rightarrow \infty}.

We establish this bound through duality. Indeed, suppose we could find non-negative coefficients {a_1,\dots,a_m} such that one had the pointwise bound

\displaystyle  \sum_{k=1}^m a_k g(kx) \leq 1 \ \ \ \ \ (1)

for all real numbers {x}. Integrating this against an arbitrary probability measure {\mu}, we would conclude

\displaystyle  (\sum_{k=1}^m a_k) \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \sum_{k=1}^m a_k \mathop{\bf E}_\mu g(kx) \leq 1

and hence

\displaystyle  c^{\mathrm{saw}}_m \leq \frac{1}{\sum_{k=1}^m a_k}.

Conversely, one can find lower bounds on {c^{\mathrm{saw}}_m} by selecting suitable candidate measures {\mu} and computing the means {\mathop{\bf E}_\mu g(kx)}. The theory of linear programming duality tells us that this method must give us the optimal bound, but one has to locate the optimal measure {\mu} and optimal weights {a_1,\dots,a_m}. This we were able to do by first doing some extensive numerics to discover these weights and measures for small values of {m}, and then doing some educated guesswork to extrapolate these examples to the general case, and then to verify the required inequalities. In case (i) the situation is particularly simple, as one can take {\mu} to be the discrete measure that assigns a probability {\frac{1}{r+2}} to the numbers {\frac{1}{2}, \frac{1}{4}, \dots, \frac{1}{2^r}} and the remaining probability of {\frac{2}{r+2}} to {\frac{1}{2^{r+1}}}, while the optimal weighted inequality (1) turns out to be

\displaystyle  2g(x) + \sum_{j=1}^r g(2^j x) \leq 1

which is easily proven by telescoping series. However the general case turned out to be significantly tricker to work out, and the verification of the optimal inequality required a delicate case analysis (reflecting the fact that equality was attained in this inequality in a large number of places).

After solving the sawtooth problem, we became interested in the analogous question for the sine function, that is to say what is the best bound {c^{\sin}_m} for the inequality

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu \sin(kx) \leq c^{\sin}_m.

The left-hand side is the smallest imaginary part of the first {m} Fourier coefficients of {\mu}. To our knowledge this quantity has not previously been studied in the Fourier analysis literature. By adopting a similar approach as for the sawtooth problem, we were able to compute this quantity exactly also:

Theorem 2 For any {m \geq 1}, one has

\displaystyle  c^{\sin}_m = \frac{m+1}{2 \sum_{1 \leq j \leq m: j \hbox{ odd}} \cot \frac{\pi j}{2m+2}}.

In particular,

\displaystyle  c^{\sin}_m = \frac{\frac{\pi}{2} + o(1)}{\log m}.

Interestingly, a closely related cotangent sum recently appeared in this MathOverflow post. Verifying the lower bound on {c^{\sin}_m} boils down to choosing the right test measure {\mu}; it turns out that one should pick the probability measure supported the {\frac{\pi j}{2m+2}} with {1 \leq j \leq m} odd, with probability proportional to {\cot \frac{\pi j}{2m+2}}, and the lower bound verification eventually follows from a classical identity

\displaystyle  \frac{m+1}{2} = \sum_{1 \leq j \leq m; j \hbox{ odd}} \cot \frac{\pi j}{2m+2} \sin \frac{\pi jk}{m+1}

for {1 \leq k \leq m}, first posed by Eisenstein in 1844 and proved by Stern in 1861. The upper bound arises from establishing the trigonometric inequality

\displaystyle  \frac{2}{(m+1)^2} \sum_{1 \leq k \leq m; k \hbox{ odd}}

\displaystyle \cot \frac{\pi k}{2m+2} ( (m+1-k) \sin kx + k \sin(m+1-k)x ) \leq 1

for all real numbers {x}, which to our knowledge is new; the left-hand side has a Fourier-analytic intepretation as convolving the Fejér kernel with a certain discretized square wave function, and this interpretation is used heavily in our proof of the inequality.

In the modern theory of higher order Fourier analysis, a key role are played by the Gowers uniformity norms {\| \|_{U^k}} for {k=1,2,\dots}. For finitely supported functions {f: {\bf Z} \rightarrow {\bf C}}, one can define the (non-normalised) Gowers norm {\|f\|_{\tilde U^k({\bf Z})}} by the formula

\displaystyle  \|f\|_{\tilde U^k({\bf Z})}^{2^k} := \sum_{n,h_1,\dots,h_k \in {\bf Z}} \prod_{\omega_1,\dots,\omega_k \in \{0,1\}} {\mathcal C}^{\omega_1+\dots+\omega_k} f(x+\omega_1 h_1 + \dots + \omega_k h_k)

where {{\mathcal C}} denotes complex conjugation, and then on any discrete interval {[N] = \{1,\dots,N\}} and any function {f: [N] \rightarrow {\bf C}} we can then define the (normalised) Gowers norm

\displaystyle  \|f\|_{U^k([N])} := \| f 1_{[N]} \|_{\tilde U^k({\bf Z})} / \|1_{[N]} \|_{\tilde U^k({\bf Z})}

where {f 1_{[N]}: {\bf Z} \rightarrow {\bf C}} is the extension of {f} by zero to all of {{\bf Z}}. Thus for instance

\displaystyle  \|f\|_{U^1([N])} = |\mathop{\bf E}_{n \in [N]} f(n)|

(which technically makes {\| \|_{U^1([N])}} a seminorm rather than a norm), and one can calculate

\displaystyle  \|f\|_{U^2([N])} \asymp (N \int_0^1 |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)|^4\ d\alpha)^{1/4} \ \ \ \ \ (1)

where {e(\theta) := e^{2\pi i \alpha}}, and we use the averaging notation {\mathop{\bf E}_{n \in A} f(n) = \frac{1}{|A|} \sum_{n \in A} f(n)}.

The significance of the Gowers norms is that they control other multilinear forms that show up in additive combinatorics. Given any polynomials {P_1,\dots,P_m: {\bf Z}^d \rightarrow {\bf Z}} and functions {f_1,\dots,f_m: [N] \rightarrow {\bf C}}, we define the multilinear form

\displaystyle  \Lambda^{P_1,\dots,P_m}(f_1,\dots,f_m) := \sum_{n \in {\bf Z}^d} \prod_{j=1}^m f_j 1_{[N]}(P_j(n)) / \sum_{n \in {\bf Z}^d} \prod_{j=1}^m 1_{[N]}(P_j(n))

(assuming that the denominator is finite and non-zero). Thus for instance

\displaystyle  \Lambda^{\mathrm{n}}(f) = \mathop{\bf E}_{n \in [N]} f(n)

\displaystyle  \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}}(f,g) = (\mathop{\bf E}_{n \in [N]} f(n)) (\mathop{\bf E}_{n \in [N]} g(n))

\displaystyle  \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+2\mathrm{r}}(f,g,h) \asymp \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [-N,N]} f(n) g(n+r) h(n+2r)

\displaystyle  \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+\mathrm{r}^2}(f,g,h) \asymp \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [-N^{1/2},N^{1/2}]} f(n) g(n+r) h(n+r^2)

where we view {\mathrm{n}, \mathrm{r}} as formal (indeterminate) variables, and {f,g,h: [N] \rightarrow {\bf C}} are understood to be extended by zero to all of {{\bf Z}}. These forms are used to count patterns in various sets; for instance, the quantity {\Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+2\mathrm{r}}(1_A,1_A,1_A)} is closely related to the number of length three arithmetic progressions contained in {A}. Let us informally say that a form {\Lambda^{P_1,\dots,P_m}(f_1,\dots,f_m)} is controlled by the {U^k[N]} norm if the form is small whenever {f_1,\dots,f_m: [N] \rightarrow {\bf C}} are {1}-bounded functions with at least one of the {f_j} small in {U^k[N]} norm. This definition was made more precise by Gowers and Wolf, who then defined the true complexity of a form {\Lambda^{P_1,\dots,P_m}} to be the least {s} such that {\Lambda^{P_1,\dots,P_m}} is controlled by the {U^{s+1}[N]} norm. For instance,
  • {\Lambda^{\mathrm{n}}} and {\Lambda^{\mathrm{n}, \mathrm{n} + \mathrm{r}}} have true complexity {0};
  • {\Lambda^{\mathrm{n}, \mathrm{n} + \mathrm{r}, \mathrm{n} + \mathrm{2r}}} has true complexity {1};
  • {\Lambda^{\mathrm{n}, \mathrm{n} + \mathrm{r}, \mathrm{n} + \mathrm{2r}, \mathrm{n} + \mathrm{3r}}} has true complexity {2};
  • The form {\Lambda^{\mathrm{n}, \mathrm{n}+2}} (which among other things could be used to count twin primes) has infinite true complexity (which is quite unfortunate for applications).
Roughly speaking, patterns of complexity {1} or less are amenable to being studied by classical Fourier analytic tools (the Hardy-Littlewood circle method); patterns of higher complexity can be handled (in principle, at least) by the methods of higher order Fourier analysis; and patterns of infinite complexity are out of range of both methods and are generally quite difficult to study. See these recent slides of myself (or this video of the lecture) for some further discussion.

Gowers and Wolf formulated a conjecture on what this complexity should be, at least for linear polynomials {P_1,\dots,P_m}; Ben Green and I thought we had resolved this conjecture back in 2010, though it turned out there was a subtle gap in our arguments and we were only able to resolve the conjecture in a partial range of cases. However, the full conjecture was recently resolved by Daniel Altman.

The {U^1} (semi-)norm is so weak that it barely controls any averages at all. For instance the average

\displaystyle  \Lambda^{2\mathrm{n}}(f) = \mathop{\bf E}_{n \in [N], \hbox{ even}} f(n)

is not controlled by the {U^1[N]} semi-norm: it is perfectly possible for a {1}-bounded function {f: [N] \rightarrow {\bf C}} to even have vanishing {U^1([N])} norm but have large value of {\Lambda^{2\mathrm{n}}(f)} (consider for instance the parity function {f(n) := (-1)^n}).

Because of this, I propose inserting an additional norm in the Gowers uniformity norm hierarchy between the {U^1} and {U^2} norms, which I will call the {U^{1^+}} (or “profinite {U^1}“) norm:

\displaystyle  \| f\|_{U^{1^+}[N]} := \frac{1}{N} \sup_P |\sum_{n \in P} f(n)| = \sup_P | \mathop{\bf E}_{n \in [N]} f 1_P(n)|

where {P} ranges over all arithmetic progressions in {[N]}. This can easily be seen to be a norm on functions {f: [N] \rightarrow {\bf C}} that controls the {U^1[N]} norm. It is also basically controlled by the {U^2[N]} norm for {1}-bounded functions {f}; indeed, if {P} is an arithmetic progression in {[N]} of some spacing {q \geq 1}, then we can write {P} as the intersection of an interval {I} with a residue class modulo {q}, and from Fourier expansion we have

\displaystyle  \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \sup_\alpha |\mathop{\bf E}_{n \in [N]} f 1_I(n) e(\alpha n)|.

If we let {\psi} be a standard bump function supported on {[-1,1]} with total mass and {\delta>0} is a parameter then

\displaystyle  \mathop{\bf E}_{n \in [N]} f 1_I(n) e(\alpha n)

\displaystyle \ll |\mathop{\bf E}_{n \in [-N,2N]; h, k \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N})

\displaystyle  1_I(n+h+k) f(n+h+k) e(\alpha(n+h+k))|

\displaystyle  \ll |\mathop{\bf E}_{n \in [-N,2N]; h, k \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+k) f(n+h+k) e(\alpha(n+h+k))|

\displaystyle + \delta

(extending {f} by zero outside of {[N]}), as can be seen by using the triangle inequality and the estimate

\displaystyle  \mathop{\bf E}_{h \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+h+k) - \mathop{\bf E}_{h \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+k)

\displaystyle \ll (1 + \mathrm{dist}(n+k, I) / \delta N)^{-2}.

After some Fourier expansion of {\delta \psi(\frac{h}{\delta N})} we now have

\displaystyle  \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \frac{1}{\delta} \sup_{\alpha,\beta} |\mathop{\bf E}_{n \in [N]; h, k \in [-N,N]} e(\beta h + \alpha (n+h+k))

\displaystyle 1_P(n+k) f(n+h+k)| + \delta.

Writing {\alpha h + \alpha(n+h+k)} as a linear combination of {n, n+h, n+k} and using the Gowers–Cauchy–Schwarz inequality, we conclude

\displaystyle  \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \frac{1}{\delta} \|f\|_{U^2([N])} + \delta

hence on optimising in {\delta} we have

\displaystyle  \| f\|_{U^{1^+}[N]} \ll \|f\|_{U^2[N]}^{1/2}.

Forms which are controlled by the {U^{1^+}} norm (but not {U^1}) would then have their true complexity adjusted to {0^+} with this insertion.

The {U^{1^+}} norm recently appeared implicitly in work of Peluse and Prendiville, who showed that the form {\Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+\mathrm{r}^2}(f,g,h)} had true complexity {0^+} in this notation (with polynomially strong bounds). [Actually, strictly speaking this control was only shown for the third function {h}; for the first two functions {f,g} one needs to localize the {U^{1^+}} norm to intervals of length {\sim \sqrt{N}}. But I will ignore this technical point to keep the exposition simple.] The weaker claim that {\Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}^2}(f,g)} has true complexity {0^+} is substantially easier to prove (one can apply the circle method together with Gauss sum estimates).

The well known inverse theorem for the {U^2} norm tells us that if a {1}-bounded function {f} has {U^2[N]} norm at least {\eta} for some {0 < \eta < 1}, then there is a Fourier phase {n \mapsto e(\alpha n)} such that

\displaystyle  |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \gg \eta^2;

this follows easily from (1) and Plancherel’s theorem. Conversely, from the Gowers–Cauchy–Schwarz inequality one has

\displaystyle  |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \ll \|f\|_{U^2[N]}.

For {U^1[N]} one has a trivial inverse theorem; by definition, the {U^1[N]} norm of {f} is at least {\eta} if and only if

\displaystyle  |\mathop{\bf E}_{n \in [N]} f(n)| \geq \eta.

Thus the frequency {\alpha} appearing in the {U^2} inverse theorem can be taken to be zero when working instead with the {U^1} norm.

For {U^{1^+}} one has the intermediate situation in which the frequency {\alpha} is not taken to be zero, but is instead major arc. Indeed, suppose that {f} is {1}-bounded with {\|f\|_{U^{1^+}[N]} \geq \eta}, thus

\displaystyle  |\mathop{\bf E}_{n \in [N]} 1_P(n) f(n)| \geq \eta

for some progression {P}. This forces the spacing {q} of this progression to be {\ll 1/\eta}. We write the above inequality as

\displaystyle  |\mathop{\bf E}_{n \in [N]} 1_{n=b\ (q)} 1_I(n) f(n)| \geq \eta

for some residue class {b\ (q)} and some interval {I}. By Fourier expansion and the triangle inequality we then have

\displaystyle  |\mathop{\bf E}_{n \in [N]} e(-an/q) 1_I(n) f(n)| \geq \eta

for some integer {a}. Convolving {1_I} by {\psi_\delta: n \mapsto \frac{1}{N\delta} \psi(\frac{n}{N\delta})} for {\delta} a small multiple of {\eta} and {\psi} a Schwartz function of unit mass with Fourier transform supported on {[-1,1]}, we have

\displaystyle  |\mathop{\bf E}_{n \in [N]} e(-an/q) (1_I * \psi_\delta)(n) f(n)| \gg \eta.

The Fourier transform {\xi \mapsto \sum_n 1_I * \psi_\delta(n) e(- \xi n)} of {1_I * \psi_\delta} is bounded by {O(N)} and supported on {[-\frac{1}{\delta N},\frac{1}{\delta N}]}, thus by Fourier expansion and the triangle inequality we have

\displaystyle  |\mathop{\bf E}_{n \in [N]} e(-an/q) e(-\xi n) f(n)| \gg \eta^2

for some {\xi \in [-\frac{1}{\delta N},\frac{1}{\delta N}]}, so in particular {\xi = O(\frac{1}{\eta N})}. Thus we have

\displaystyle  |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \gg \eta^2 \ \ \ \ \ (2)

for some {\alpha} of the major arc form {\alpha = \frac{a}{q} + O(1/\eta)} with {1 \leq q \leq 1/\eta}. Conversely, for {\alpha} of this form, some routine summation by parts gives the bound

\displaystyle  |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \ll \frac{q}{\eta} \|f\|_{U^{1^+}[N]} \ll \frac{1}{\eta^2} \|f\|_{U^{1^+}[N]}

so if (2) holds for a {1}-bounded {f} then one must have {\|f\|_{U^{1^+}[N]} \gg \eta^4}.

Here is a diagram showing some of the control relationships between various Gowers norms, multilinear forms, and duals of classes {{\mathcal F}} of functions (where each class of functions {{\mathcal F}} induces a dual norm {\| f \|_{{\mathcal F}^*} := \sup_{\phi \in {\mathcal F}} \mathop{\bf E}_{n \in[N]} f(n) \overline{\phi(n)}}:

Here I have included the three classes of functions that one can choose from for the {U^3} inverse theorem, namely degree two nilsequences, bracket quadratic phases, and local quadratic phases, as well as the more narrow class of globally quadratic phases.

The Gowers norms have counterparts for measure-preserving systems {(X,T,\mu)}, known as Host-Kra seminorms. The {U^1(X)} norm can be defined for {f \in L^\infty(X)} as

\displaystyle  \|f\|_{U^1(X)} := \lim_{N \rightarrow \infty} \int_X |\mathop{\bf E}_{n \in [N]} T^n f|\ d\mu

and the {U^2} norm can be defined as

\displaystyle  \|f\|_{U^2(X)}^4 := \lim_{N \rightarrow \infty} \mathop{\bf E}_{n \in [N]} \| T^n f \overline{f} \|_{U^1(X)}^2.

The {U^1(X)} seminorm is orthogonal to the invariant factor {Z^0(X)} (generated by the (almost everywhere) invariant measurable subsets of {X}) in the sense that a function {f \in L^\infty(X)} has vanishing {U^1(X)} seminorm if and only if it is orthogonal to all {Z^0(X)}-measurable (bounded) functions. Similarly, the {U^2(X)} norm is orthogonal to the Kronecker factor {Z^1(X)}, generated by the eigenfunctions of {X} (that is to say, those {f} obeying an identity {Tf = \lambda f} for some {T}-invariant {\lambda}); for ergodic systems, it is the largest factor isomorphic to rotation on a compact abelian group. In analogy to the Gowers {U^{1^+}[N]} norm, one can then define the Host-Kra {U^{1^+}(X)} seminorm by

\displaystyle  \|f\|_{U^{1^+}(X)} := \sup_{q \geq 1} \frac{1}{q} \lim_{N \rightarrow \infty} \int_X |\mathop{\bf E}_{n \in [N]} T^{qn} f|\ d\mu;

it is orthogonal to the profinite factor {Z^{0^+}(X)}, generated by the periodic sets of {X} (or equivalently, by those eigenfunctions whose eigenvalue is a root of unity); for ergodic systems, it is the largest factor isomorphic to rotation on a profinite abelian group.

Joni Teräväinen and myself have just uploaded to the arXiv our preprint “Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions“. This paper makes quantitative the Gowers uniformity estimates on the Möbius function {\mu} and the von Mangoldt function {\Lambda}.

To discuss the results we first discuss the situation of the Möbius function, which is technically simpler in some (though not all) ways. We assume familiarity with Gowers norms and standard notations around these norms, such as the averaging notation {\mathop{\bf E}_{n \in [N]}} and the exponential notation {e(\theta) = e^{2\pi i \theta}}. The prime number theorem in qualitative form asserts that

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) = o(1)

as {N \rightarrow \infty}. With Vinogradov-Korobov error term, the prime number theorem is strengthened to

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} );

we refer to such decay bounds (With {\exp(-c\log^c N)} type factors) as pseudopolynomial decay. Equivalently, we obtain pseudopolynomial decay of Gowers {U^1} seminorm of {\mu}:

\displaystyle  \| \mu \|_{U^1([N])} \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} ).

As is well known, the Riemann hypothesis would be equivalent to an upgrade of this estimate to polynomial decay of the form

\displaystyle  \| \mu \|_{U^1([N])} \ll_\varepsilon N^{-1/2+\varepsilon}

for any {\varepsilon>0}.

Once one restricts to arithmetic progressions, the situation gets worse: the Siegel-Walfisz theorem gives the bound

\displaystyle  \| \mu 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll_A \log^{-A} N \ \ \ \ \ (1)

for any residue class {a \hbox{ mod } q} and any {A>0}, but with the catch that the implied constant is ineffective in {A}. This ineffectivity cannot be removed without further progress on the notorious Siegel zero problem.

In 1937, Davenport was able to show the discorrelation estimate

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) e(-\alpha n) \ll_A \log^{-A} N

for any {A>0} uniformly in {\alpha \in {\bf R}}, which leads (by standard Fourier arguments) to the Fourier uniformity estimate

\displaystyle  \| \mu \|_{U^2([N])} \ll_A \log^{-A} N.

Again, the implied constant is ineffective. If one insists on effective constants, the best bound currently available is

\displaystyle  \| \mu \|_{U^2([N])} \ll \log^{-c} N \ \ \ \ \ (2)

for some small effective constant {c>0}.

For the situation with the {U^3} norm the previously known results were much weaker. Ben Green and I showed that

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \ll_{A,F,G/\Gamma} \log^{-A} N \ \ \ \ \ (3)

uniformly for any {A>0}, any degree two (filtered) nilmanifold {G/\Gamma}, any polynomial sequence {g: {\bf Z} \rightarrow G}, and any Lipschitz function {F}; again, the implied constants are ineffective. On the other hand, in a separate paper of Ben Green and myself, we established the following inverse theorem: if for instance we knew that

\displaystyle  \| \mu \|_{U^3([N])} \geq \delta

for some {0 < \delta < 1/2}, then there exists a degree two nilmanifold {G/\Gamma} of dimension {O( \delta^{-O(1)} )}, complexity {O( \delta^{-O(1)} )}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and Lipschitz function {F} of Lipschitz constant {O(\delta^{-O(1)})} such that

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \gg \exp(-\delta^{-O(1)}).

Putting the two assertions together and comparing all the dependencies on parameters, one can establish the qualitative decay bound

\displaystyle  \| \mu \|_{U^3([N])} = o(1).

However the decay rate {o(1)} produced by this argument is completely ineffective: obtaining a bound on when this {o(1)} quantity dips below a given threshold {\delta} depends on the implied constant in (3) for some {G/\Gamma} whose dimension depends on {\delta}, and the dependence on {\delta} obtained in this fashion is ineffective in the face of a Siegel zero.

For higher norms {U^k, k \geq 3}, the situation is even worse, because the quantitative inverse theory for these norms is poorer, and indeed it was only with the recent work of Manners that any such bound is available at all (at least for {k>4}). Basically, Manners establishes if

\displaystyle  \| \mu \|_{U^k([N])} \geq \delta

then there exists a degree {k-1} nilmanifold {G/\Gamma} of dimension {O( \delta^{-O(1)} )}, complexity {O( \exp\exp(\delta^{-O(1)}) )}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and Lipschitz function {F} of Lipschitz constant {O(\exp\exp(\delta^{-O(1)}))} such that

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \gg \exp\exp(-\delta^{-O(1)}).

(We allow all implied constants to depend on {k}.) Meanwhile, the bound (3) was extended to arbitrary nilmanifolds by Ben and myself. Again, the two results when concatenated give the qualitative decay

\displaystyle  \| \mu \|_{U^k([N])} = o(1)

but the decay rate is completely ineffective.

Our first result gives an effective decay bound:

Theorem 1 For any {k \geq 2}, we have {\| \mu \|_{U^k([N])} \ll (\log\log N)^{-c_k}} for some {c_k>0}. The implied constants are effective.

This is off by a logarithm from the best effective bound (2) in the {k=2} case. In the {k=3} case there is some hope to remove this logarithm based on the improved quantitative inverse theory currently available in this case, but there is a technical obstruction to doing so which we will discuss later in this post. For {k>3} the above bound is the best one could hope to achieve purely using the quantitative inverse theory of Manners.

We have analogues of all the above results for the von Mangoldt function {\Lambda}. Here a complication arises that {\Lambda} does not have mean close to zero, and one has to subtract off some suitable approximant {\Lambda^\sharp} to {\Lambda} before one would expect good Gowers norms bounds. For the prime number theorem one can just use the approximant {1}, giving

\displaystyle  \| \Lambda - 1 \|_{U^1([N])} \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} )

but even for the prime number theorem in arithmetic progressions one needs a more accurate approximant. In our paper it is convenient to use the “Cramér approximant”

\displaystyle  \Lambda_{\hbox{Cram\'er}}(n) := \frac{W}{\phi(W)} 1_{(n,W)=1}

where

\displaystyle  W := \prod_{p<Q} p

and {Q} is the quasipolynomial quantity

\displaystyle  Q = \exp(\log^{1/10} N). \ \ \ \ \ (4)

Then one can show from the Siegel-Walfisz theorem and standard bilinear sum methods that

\displaystyle  \mathop{\bf E}_{n \in [N]} (\Lambda - \Lambda_{\hbox{Cram\'er}}(n)) e(-\alpha n) \ll_A \log^{-A} N

and

\displaystyle  \| \Lambda - \Lambda_{\hbox{Cram\'er}}\|_{U^2([N])} \ll_A \log^{-A} N

for all {A>0} and {\alpha \in {\bf R}} (with an ineffective dependence on {A}), again regaining effectivity if {A} is replaced by a sufficiently small constant {c>0}. All the previously stated discorrelation and Gowers uniformity results for {\mu} then have analogues for {\Lambda}, and our main result is similarly analogous:

Theorem 2 For any {k \geq 2}, we have {\| \Lambda - \Lambda_{\hbox{Cram\'er}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}} for some {c_k>0}. The implied constants are effective.

By standard methods, this result also gives quantitative asymptotics for counting solutions to various systems of linear equations in primes, with error terms that gain a factor of {O((\log\log N)^{-c})} with respect to the main term.

We now discuss the methods of proof, focusing first on the case of the Möbius function. Suppose first that there is no “Siegel zero”, by which we mean a quadratic character {\chi} of some conductor {q \leq Q} with a zero {L(\beta,\chi)} with {1 - \beta \leq \frac{c}{\log Q}} for some small absolute constant {c>0}. In this case the Siegel-Walfisz bound (1) improves to a quasipolynomial bound

\displaystyle  \| \mu 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N). \ \ \ \ \ (5)

To establish Theorem 1 in this case, it suffices by Manners’ inverse theorem to establish the polylogarithmic bound

\displaystyle  \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N) \ \ \ \ \ (6)

for all degree {k-1} nilmanifolds {G/\Gamma} of dimension {O((\log\log N)^c)} and complexity {O( \exp(\log^c N))}, all polynomial sequences {g}, and all Lipschitz functions {F} of norm {O( \exp(\log^c N))}. If the nilmanifold {G/\Gamma} had bounded dimension, then one could repeat the arguments of Ben and myself more or less verbatim to establish this claim from (5), which relied on the quantitative equidistribution theory on nilmanifolds developed in a separate paper of Ben and myself. Unfortunately, in the latter paper the dependence of the quantitative bounds on the dimension {d} was not explicitly given. In an appendix to the current paper, we go through that paper to account for this dependence, showing that all exponents depend at most doubly exponentially in the dimension {d}, which is barely sufficient to handle the dimension of {O((\log\log N)^c)} that arises here.

Now suppose we have a Siegel zero {L(\beta,\chi)}. In this case the bound (5) will not hold in general, and hence also (6) will not hold either. Here, the usual way out (while still maintaining effective estimates) is to approximate {\mu} not by {0}, but rather by a more complicated approximant {\mu_{\hbox{Siegel}}} that takes the Siegel zero into account, and in particular is such that one has the (effective) pseudopolynomial bound

\displaystyle  \| (\mu - \mu_{\hbox{Siegel}}) 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N) \ \ \ \ \ (7)

for all residue classes {a \hbox{ mod } q}. The Siegel approximant to {\mu} is actually a little bit complicated, and to our knowledge the first appearance of this sort of approximant only appears as late as this 2010 paper of Germán and Katai. Our version of this approximant is defined as the multiplicative function such that

\displaystyle \mu_{\hbox{Siegel}}(p^j) = \mu(p^j)

when {p < Q}, and

\displaystyle  \mu_{\hbox{Siegel}}(n) = \alpha n^{\beta-1} \chi(n)

when {n} is coprime to all primes {p<Q}, and {\alpha} is a normalising constant given by the formula

\displaystyle  \alpha := \frac{1}{L'(\beta,\chi)} \prod_{p<Q} (1-\frac{1}{p})^{-1} (1 - \frac{\chi(p)}{p^\beta})^{-1}

(this constant ends up being of size {O(1)} and plays only a minor role in the analysis). This is a rather complicated formula, but it seems to be virtually the only choice of approximant that allows for bounds such as (7) to hold. (This is the one aspect of the problem where the von Mangoldt theory is simpler than the Möbius theory, as in the former one only needs to work with very rough numbers for which one does not need to make any special accommodations for the behavior at small primes when introducing the Siegel correction term.) With this starting point it is then possible to repeat the analysis of my previous papers with Ben and obtain the pseudopolynomial discorrelation bound

\displaystyle  \mathop{\bf E}_{n \in [N]} (\mu - \mu_{\hbox{Siegel}})(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N)

for {F(g(n)\Gamma)} as before, which when combined with Manners’ inverse theorem gives the doubly logarithmic bound

\displaystyle \| \mu - \mu_{\hbox{Siegel}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}.

Meanwhile, a direct sieve-theoretic computation ends up giving the singly logarithmic bound

\displaystyle \| \mu_{\hbox{Siegel}} \|_{U^k([N])} \ll \log^{-c_k} N

(indeed, there is a good chance that one could improve the bounds even further, though it is not helpful for this current argument to do so). Theorem 1 then follows from the triangle inequality for the Gowers norm. It is interesting that the Siegel approximant {\mu_{\hbox{Siegel}}} seems to play a rather essential component in the proof, even if it is absent in the final statement. We note that this approximant seems to be a useful tool to explore the “illusory world” of the Siegel zero further; see for instance the recent paper of Chinis for some work in this direction.

For the analogous problem with the von Mangoldt function (assuming a Siegel zero for sake of discussion), the approximant {\Lambda_{\hbox{Siegel}}} is simpler; we ended up using

\displaystyle \Lambda_{\hbox{Siegel}}(n) = \Lambda_{\hbox{Cram\'er}}(n) (1 - n^{\beta-1} \chi(n))

which allows one to state the standard prime number theorem in arithmetic progressions with classical error term and Siegel zero term compactly as

\displaystyle  \| (\Lambda - \Lambda_{\hbox{Siegel}}) 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N).

Routine modifications of previous arguments also give

\displaystyle  \mathop{\bf E}_{n \in [N]} (\Lambda - \Lambda_{\hbox{Siegel}})(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N) \ \ \ \ \ (8)

and

\displaystyle \| \Lambda_{\hbox{Siegel}} \|_{U^k([N])} \ll \log^{-c_k} N.

The one tricky new step is getting from the discorrelation estimate (8) to the Gowers uniformity estimate

\displaystyle \| \Lambda - \Lambda_{\hbox{Siegel}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}.

One cannot directly apply Manners’ inverse theorem here because {\Lambda} and {\Lambda_{\hbox{Siegel}}} are unbounded. There is a standard tool for getting around this issue, now known as the dense model theorem, which is the standard engine powering the transference principle from theorems about bounded functions to theorems about certain types of unbounded functions. However the quantitative versions of the dense model theorem in the literature are expensive and would basically weaken the doubly logarithmic gain here to a triply logarithmic one. Instead, we bypass the dense model theorem and directly transfer the inverse theorem for bounded functions to an inverse theorem for unbounded functions by using the densification approach to transference introduced by Conlon, Fox, and Zhao. This technique turns out to be quantitatively quite efficient (the dependencies of the main parameters in the transference are polynomial in nature), and also has the technical advantage of avoiding the somewhat tricky “correlation condition” present in early transference results which are also not beneficial for quantitative bounds.

In principle, the above results can be improved for {k=3} due to the stronger quantitative inverse theorems in the {U^3} setting. However, there is a bottleneck that prevents us from achieving this, namely that the equidistribution theory of two-step nilmanifolds has exponents which are exponential in the dimension rather than polynomial in the dimension, and as a consequence we were unable to improve upon the doubly logarithmic results. Specifically, if one is given a sequence of bracket quadratics such as {\lfloor \alpha_1 n \rfloor \beta_1 n, \dots, \lfloor \alpha_d n \rfloor \beta_d n} that fails to be {\delta}-equidistributed, one would need to establish a nontrivial linear relationship modulo 1 between the {\alpha_1,\beta_1,\dots,\alpha_d,\beta_d} (up to errors of {O(1/N)}), where the coefficients are of size {O(\delta^{-d^{O(1)}})}; current methods only give coefficient bounds of the form {O(\delta^{-\exp(d^{O(1)})})}. An old result of Schmidt demonstrates proof of concept that these sorts of polynomial dependencies on exponents is possible in principle, but actually implementing Schmidt’s methods here seems to be a quite non-trivial task. There is also another possible route to removing a logarithm, which is to strengthen the inverse {U^3} theorem to make the dimension of the nilmanifold logarithmic in the uniformity parameter {\delta} rather than polynomial. Again, the Freiman-Bilu theorem (see for instance this paper of Ben and myself) demonstrates proof of concept that such an improvement in dimension is possible, but some work would be needed to implement it.

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