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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
The conjecture gets more difficult as increases, and also becomes more difficult the more slowly
grows with
. The
conjecture is equivalent to the assertion
For , the conjecture is equivalent to the assertion
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
We give two applications of this higher order Fourier uniformity. One regards the growth of the number
The second application is to obtain cancellation for various polynomial averages involving the Liouville function or von Mangoldt function
, such as
Joni Teräväinen and I have just uploaded to the arXiv our paper “Value patterns of multiplicative functions and related sequences“, submitted to Forum of Mathematics, Sigma. This paper explores how to use recent technology on correlations of multiplicative (or nearly multiplicative functions), such as the “entropy decrement method”, in conjunction with techniques from additive combinatorics, to establish new results on the sign patterns of functions such as the Liouville function . For instance, with regards to length 5 sign patterns
of the Liouville function, we can now show that at least of the
possible sign patterns in
occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately
seems to be the limitation of our methods.)
The Liouville function can be written as , where
is the number of prime factors of
(counting multiplicity). One can also consider the variant
, which is a completely multiplicative function taking values in the cube roots of unity
. Here we are able to show that all
sign patterns in
occur with positive lower density as sign patterns
of this function. The analogous result for
was already known (see this paper of Matomäki, Radziwiłł, and myself), and in that case it is even known that all sign patterns occur with equal logarithmic density
(from this paper of myself and Teräväinen), but these techniques barely fail to handle the
case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the
case) and an additional additive combinatorial tool is needed. After applying existing technology (such as entropy decrement methods), the problem roughly speaking reduces to locating patterns
for a certain partition
of a compact abelian group
(think for instance of the unit circle
, although the general case is a bit more complicated, in particular if
is disconnected then there is a certain “coprimality” constraint on
, also we can allow the
to be replaced by any
with
divisible by
), with each of the
having measure
. An inequality of Kneser just barely fails to guarantee the existence of such patterns, but by using an inverse theorem for Kneser’s inequality in this previous paper of mine we are able to identify precisely the obstruction for this method to work, and rule it out by an ad hoc method.
The same techniques turn out to also make progress on some conjectures of Erdös-Pomerance and Hildebrand regarding patterns of the largest prime factor of a natural number
. For instance, we improve results of Erdös-Pomerance and of Balog demonstrating that the inequalities
and
each hold for infinitely many , by demonstrating the stronger claims that the inequalities
and
each hold for a set of of positive lower density. As a variant, we also show that we can find a positive density set of
for which
for any fixed (this improves on a previous result of Hildebrand with
replaced by
. A number of other results of this type are also obtained in this paper.
In order to obtain these sorts of results, one needs to extend the entropy decrement technology from the setting of multiplicative functions to that of what we call “weakly stable sets” – sets which have some multiplicative structure, in the sense that (roughly speaking) there is a set
such that for all small primes
, the statements
and
are roughly equivalent to each other. For instance, if
is a level set
, one would take
; if instead
is a set of the form
, then one can take
. When one has such a situation, then very roughly speaking, the entropy decrement argument then allows one to estimate a one-parameter correlation such as
with a two-parameter correlation such as
(where we will be deliberately vague as to how we are averaging over and
), and then the use of the “linear equations in primes” technology of Ben Green, Tamar Ziegler, and myself then allows one to replace this average in turn by something like
where is constrained to be not divisible by small primes but is otherwise quite arbitrary. This latter average can then be attacked by tools from additive combinatorics, such as translation to a continuous group model (using for instance the Furstenberg correspondence principle) followed by tools such as Kneser’s inequality (or inverse theorems to that inequality).
Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures“. This is a sequel to our previous paper that studied logarithmic correlations of the form
where were bounded multiplicative functions,
were fixed shifts,
was a quantity going off to infinity, and
was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions
. Furthermore, if
(weakly) pretended to be a Dirichlet character
, then the
could be chosen to be
–isotypic in the sense that
whenever
are integers with
coprime to the periods of
and
; otherwise, if
did not weakly pretend to be any Dirichlet character
, then
vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.
The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages
where is an additional parameter. Our main result was now as follows. If
did not weakly pretend to be a twisted Dirichlet character
, then
converged to zero on (doubly logarithmic) average as
. If instead
did pretend to be such a twisted Dirichlet character, then
converged on (doubly logarithmic) average to a limit
of
-isotypic functions
. Thus, roughly speaking, one has the approximation
for most .
Informally, this says that at almost all scales (where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character
(which interacts with the Archimedean parameter
in much the same way that the Dirichlet character
interacts with the non-Archimedean parameter
). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales
of logarithmic density zero. For instance, the Chowla conjecture
is now established for either odd or equal to
, so long as one excludes an exceptional set of scales.
In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on . The first, coming from recent results in ergodic theory, was to show that
was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form
for “most” primes and integers
. Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.
In the current situation, each is approximated by a nilsequence, but the nilsequence can vary with
(although there is some useful “Lipschitz continuity” of this nilsequence with respect to the
parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form
for “most” . The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter
is the following cohomological statement: if one has a map
that was a quasimorphism in the sense that
for all
and some small
, then there exists a real number
such that
for all small
. This is achieved by applying a standard “cocycle averaging argument” to the cocycle
.
It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.
Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic
for all odd and all integers
(that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function
); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the
level, which is unfortunate since the first non-trivial odd case
requires quantitative control on the
level. (But it may be possible to make the Gowers uniformity bounds for
quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)
Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that
whenever were sequences going to infinity,
were distinct integers, and
were
-bounded multiplicative functions which were non-pretentious in the sense that
for all Dirichlet characters and for
. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture
for fixed any non-zero , where
was the Liouville function.
One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that
for all odd and all integers
(which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument
).
For the more general Elliott conjecture, we can show that
for any , any integers
and any bounded multiplicative functions
, unless the product
weakly pretends to be a Dirichlet character
in the sense that
This can be seen to imply (2) as a special case. Even when does pretend to be a Dirichlet character
, we can still say something: if the limits
exist for each (which can be guaranteed if we pass to a suitable subsequence), then
is the uniform limit of periodic functions
, each of which is
–isotypic in the sense that
whenever
are integers with
coprime to the periods of
and
. This does not pin down the value of any single correlation
, but does put significant constraints on how these correlations may vary with
.
Among other things, this allows us to show that all possible length four sign patterns
of the Liouville function occur with positive density, and all
possible length four sign patterns
occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)
To describe the argument, let us focus for simplicity on the case of the Liouville correlations
assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime
and observe that
for any
, which allows us to rewrite (3) as
Making the change of variables , we obtain
The difference between and
is negligible in the limit (here is where we crucially rely on the log-averaging), hence
and thus by (3) we have
The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors
behave like independent random variables as
varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the
factors). We thus obtain the approximate isotopy property
for most and
.
On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation
for some probability space equipped with a measure-preserving invertible map
. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form
where is a nilsequence, and
goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on
, which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on
so that one still has good control when restricting to primes, or constant multiples of primes.
Ignoring the small error , we can now combine (5) to conclude that
Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece
and an “irrational” or “minor arc” piece
. The contribution of the minor arc piece
can be shown to mostly cancel itself out after dilating by primes
and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with
which already shows (heuristically, at least) the claim that can be approximated by periodic functions
which are isotopic in the sense that
But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes
that are
modulo the period of
, and conclude now that
vanishes identically, which (heuristically, at least) gives (2).
The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “
-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form
where ranges over a large range of integers coprime to some primorial
. On the other hand, by iterating (4) we have
for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form
For odd, one can combine the two approximations to conclude that
. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)
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