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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

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, and ${\alpha}$ is a normalising constant given by the formula

$\displaystyle \alpha := \frac{1}{L'(\beta,\chi)} \prod_{p

(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.