Higher uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki, Maksym Radziwill, Joni Teräväinen, Tamar Ziegler and I have uploaded to the arXiv our paper Higher uniformity of bounded multiplicative functions in short intervals on average. This paper (which originated from a working group at an AIM workshop on Sarnak’s conjecture) focuses on the local Fourier uniformity conjecture for bounded multiplicative functions such as the Liouville function . One form of this conjecture is the assertion that

as for any fixed and any that goes to infinity as , where is the (normalized) Gowers uniformity norm. Among other things this conjecture implies (logarithmically averaged version of) the Chowla and Sarnak conjectures for the Liouville function (or the Möbius function), see this previous blog post.

The conjecture gets more difficult as increases, and also becomes more difficult the more slowly grows with . The conjecture is equivalent to the assertion

which was proven (for arbitrarily slowly growing ) in a landmark paper of Matomäki and Radziwill, discussed for instance in this blog post.

For , the conjecture is equivalent to the assertion

This remains open for sufficiently slowly growing (and it would be a major breakthrough in particular if one could obtain this bound for as small as for any fixed , particularly if applicable to more general bounded multiplicative functions than , as this would have new implications for a generalization of the Chowla conjecture known as the Elliott conjecture). Recently, Kaisa, Maks and myself were able to establish this conjecture in the range (in fact we have since worked out in the current paper that we can get as small as ). In our current paper we establish Fourier uniformity conjecture for higher for the same range of . This in particular implies local orthogonality to polynomial phases,

where denotes the polynomials of degree at most , but the full conjecture is a bit stronger than this, establishing the more general statement

for any degree filtered nilmanifold and Lipschitz function , where now ranges over polynomial maps from to . The method of proof follows the same general strategy as in the previous paper with Kaisa and Maks. (The equivalence of (4) and (1) follows from the inverse conjecture for the Gowers norms, proven in this paper.) We quickly sketch first the proof of (3), using very informal language to avoid many technicalities regarding the precise quantitative form of various estimates. If the estimate (3) fails, then we have the correlation estimate

for many and some polynomial depending on . The difficulty here is to understand how can depend on . We write the above correlation estimate more suggestively as

Because of the multiplicativity at small primes , one expects to have a relation of the form

for many for which for some small primes . (This can be formalised using an inequality of Elliott related to the Turan-Kubilius theorem.) This gives a relationship between and for “edges” in a rather sparse “graph” connecting the elements of say . Using some graph theory one can locate some non-trivial “cycles” in this graph that eventually lead (in conjunction to a certain technical but important “Chinese remainder theorem” step to modify the to eliminate a rather serious “aliasing” issue that was already discussed in this previous post) to obtain functional equations of the form

for some large and close (but not identical) integers , where should be viewed as a first approximation (ignoring a certain “profinite” or “major arc” term for simplicity) as “differing by a slowly varying polynomial” and the polynomials should now be viewed as taking values on the reals rather than the integers. This functional equation can be solved to obtain a relation of the form

for some real number of polynomial size, and with further analysis of the relation (5) one can make basically independent of . This simplifies (3) to something like

and this is now of a form that can be treated by the theorem of Matomäki and Radziwill (because is a bounded multiplicative function). (Actually because of the profinite term mentioned previously, one also has to insert a Dirichlet character of bounded conductor into this latter conclusion, but we will ignore this technicality.)

Now we apply the same strategy to (4). For abelian the claim follows easily from (3), so we focus on the non-abelian case. One now has a polynomial sequence attached to many , and after a somewhat complicated adaptation of the above arguments one again ends up with an approximate functional equation

where the relation is rather technical and will not be detailed here. A new difficulty arises in that there are some unwanted solutions to this equation, such as

for some , which do not necessarily lead to multiplicative characters like as in the polynomial case, but instead to some unfriendly looking “generalized multiplicative characters” (think of as a rough caricature). To avoid this problem, we rework the graph theory portion of the argument to produce not just one functional equation of the form (6)for each , but many, leading to dilation invariances

for a “dense” set of . From a certain amount of Lie algebra theory (ultimately arising from an understanding of the behaviour of the exponential map on nilpotent matrices, and exploiting the hypothesis that is non-abelian) one can conclude that (after some initial preparations to avoid degenerate cases) must behave like for some central element of . This eventually brings one back to the multiplicative characters that arose in the polynomial case, and the arguments now proceed as before.

We give two applications of this higher order Fourier uniformity. One regards the growth of the number

of length sign patterns in the Liouville function. The Chowla conjecture implies that , but even the weaker conjecture of Sarnak that for some remains open. Until recently, the best asymptotic lower bound on was , due to McNamara; with our result, we can now show for any (in fact we can get for any ). The idea is to repeat the now-standard argument to exploit multiplicativity at small primes to deduce Chowla-type conjectures from Fourier uniformity conjectures, noting that the Chowla conjecture would give all the sign patterns one could hope for. The usual argument here uses the “entropy decrement argument” to eliminate a certain error term (involving the large but mean zero factor ). However the observation is that if there are extremely few sign patterns of length , then the entropy decrement argument is unnecessary (there isn’t much entropy to begin with), and a more low-tech moment method argument (similar to the derivation of Chowla’s conjecture from Sarnak’s conjecture, as discussed for instance in this post) gives enough of Chowla’s conjecture to produce plenty of length sign patterns. If there are not extremely few sign patterns of length then we are done anyway. One quirk of this argument is that the sign patterns it produces may only appear exactly once; in contrast with preceding arguments, we were not able to produce a large number of sign patterns that each occur infinitely often.

The second application is to obtain cancellation for various polynomial averages involving the Liouville function or von Mangoldt function , such as

or

where are polynomials of degree at most , no two of which differ by a constant (the latter is essential to avoid having to establish the Chowla or Hardy-Littlewood conjectures, which of course remain open). Results of this type were previously obtained by Tamar Ziegler and myself in the “true complexity zero” case when the polynomials had distinct degrees, in which one could use the theory of Matomäki and Radziwill; now that higher is available at the scale we can now remove this restriction.

The sunflower lemma via Shannon entropy

A family of sets for some is a sunflower if there is a core set contained in each of the such that the petal sets are disjoint. If , let denote the smallest natural number with the property that any family of distinct sets of cardinality at most contains distinct elements that form a sunflower. The celebrated Erdös-Rado theorem asserts that is finite; in fact Erdös and Rado gave the bounds

The sunflower conjecture asserts in fact that the upper bound can be improved to . This remains open at present despite much effort (including a Polymath project); after a long series of improvements to the upper bound, the best general bound known currently is

for all , established in 2019 by Rao (building upon a recent breakthrough a month previously of Alweiss, Lovett, Wu, and Zhang). Here we remove the easy cases or in order to make the logarithmic factor a little cleaner.

Rao’s argument used the Shannon noiseless coding theorem. It turns out that the argument can be arranged in the very slightly different language of Shannon entropy, and I would like to present it here. The argument proceeds by locating the core and petals of the sunflower separately (this strategy is also followed in Alweiss-Lovett-Wu-Zhang). In both cases the following definition will be key. In this post all random variables, such as random sets, will be understood to be discrete random variables taking values in a finite range. We always use boldface symbols to denote random variables, and non-boldface for deterministic quantities.

Definition 1 (Spread set) Let . A random set is said to be -spread if one has

for all sets . A family of sets is said to be -spread if is non-empty and the random variable is -spread, where is drawn uniformly from .

The core can then be selected greedily in such a way that the remainder of a family becomes spread:

Lemma 2 (Locating the core) Let be a family of subsets of a finite set , each of cardinality at most , and let . Then there exists a “core” set of cardinality at most such that the set

has cardinality at least , and such that the family is -spread. Furthermore, if and the are distinct, then .

Proof: We may assume is non-empty, as the claim is trivial otherwise. For any , define the quantity

and let be a subset of that maximizes . Since and when , we see that . If the are distinct and , then we also have when , thus in this case we have .

Let be the set (3). Since , is non-empty. It remains to check that the family is -spread. But for any and drawn uniformly at random from one has

Observe that , and the probability is only non-empty when are disjoint, so that . The claim follows.

In view of the above lemma, the bound (2) will then follow from

Proposition 3 (Locating the petals) Let be natural numbers, and suppose that for a sufficiently large constant . Let be a finite family of subsets of a finite set , each of cardinality at most which is -spread. Then there exist such that is disjoint.

Indeed, to prove (2), we assume that is a family of sets of cardinality greater than for some ; by discarding redundant elements and sets we may assume that is finite and that all the are contained in a common finite set . Apply Lemma 2 to find a set of cardinality such that the family is -spread. By Proposition 3 we can find such that are disjoint; since these sets have cardinality , this implies that the are distinct. Hence form a sunflower as required.

Remark 4 Proposition 3 is easy to prove if we strengthen the condition on to . In this case, we have for every , hence by the union bound we see that for any with there exists such that is disjoint from the set , which has cardinality at most . Iterating this, we obtain the conclusion of Proposition 3 in this case. This recovers a bound of the form , and by pursuing this idea a little further one can recover the original upper bound (1) of Erdös and Rado.

It remains to prove Proposition 3. In fact we can locate the petals one at a time, placing each petal inside a random set.

Proposition 5 (Locating a single petal) Let the notation and hypotheses be as in Proposition 3. Let be a random subset of , such that each lies in with an independent probability of . Then with probability greater than , contains one of the .

To see that Proposition 5 implies Proposition 3, we randomly partition into by placing each into one of the , chosen uniformly and independently at random. By Proposition 5 and the union bound, we see that with positive probability, it is simultaneously true for all that each contains one of the . Selecting one such for each , we obtain the required disjoint petals.

We will prove Proposition 5 by gradually increasing the density of the random set and arranging the sets to get quickly absorbed by this random set. The key iteration step is

Proposition 6 (Refinement inequality) Let and . Let be a random subset of a finite set which is -spread, and let be a random subset of independent of , such that each lies in with an independent probability of . Then there exists another -spread random subset of whose support is contained in the support of , such that and

Note that a direct application of the first moment method gives only the bound

but the point is that by switching from to an equivalent we can replace the factor by a quantity significantly smaller than .

One can iterate the above proposition, repeatedly replacing with (noting that this preserves the -spread nature of ) to conclude

Corollary 7 (Iterated refinement inequality) Let , , and . Let be a random subset of a finite set which is -spread, and let be a random subset of independent of , such that each lies in with an independent probability of . Then there exists another random subset of with support contained in the support of , such that

Now we can prove Proposition 5. Let be chosen shortly. Applying Corollary 7 with drawn uniformly at random from the , and setting , or equivalently , we have

In particular, if we set , so that , then by choice of we have , hence

In particular with probability at least , there must exist such that , giving the proposition.

It remains to establish Proposition 6. This is the difficult step, and requires a clever way to find the variant of that has better containment properties in than does. The main trick is to make a conditional copy of that is conditionally independent of subject to the constraint . The point here is that this constrant implies the inclusions

and

Because of the -spread hypothesis, it is hard for to contain any fixed large set. If we could apply this observation in the contrapositive to we could hope to get a good upper bound on the size of and hence on thanks to (4). One can also hope to improve such an upper bound by also employing (5), since it is also hard for the random set to contain a fixed large set. There are however difficulties with implementing this approach due to the fact that the random sets are coupled with in a moderately complicated fashion. In Rao’s argument a somewhat complicated encoding scheme was created to give information-theoretic control on these random variables; below the fold we accomplish a similar effect by using Shannon entropy inequalities in place of explicit encoding. A certain amount of information-theoretic sleight of hand is required to decouple certain random variables to the extent that the Shannon inequalities can be effectively applied. The argument bears some resemblance to the “entropy compression method” discussed in this previous blog post; there may be a way to more explicitly express the argument below in terms of that method. (There is also some kinship with the method of dependent random choice, which is used for instance to establish the Balog-Szemerédi-Gowers lemma, and was also translated into information theoretic language in these unpublished notes of Van Vu and myself.)