Updates on my research và expository papers, discussion of open problems, và other maths-related topics. By Terence Tao


Higher uniformity of arithmetic functions in short intervals I. Allintervals


Kaisa Matomäki, Xuancheng Shao, Joni Teräväinen, và myself have just uploaded to the arXiv our preprint “Higher uniformity of arithmetic functions in short intervals I. All intervals“. This paper investigates the higher order (Gowers) uniformity of standard arithmetic functions in analytic number theory (and specifically, the Möbius function

*
, the von Mangoldt function
*
, & the generalised divisor functions
*
) in short intervals
*
}" class="latex" />, where
*
is large &
*
lies in the range
*
for a fixed constant
*
was “major arc”, together with an error term. We found it convenient to cancel off such main terms by subtracting an approximant
*
from each of the arithmetic functions
*
& then getting upper bounds on remainder correlations such as

*
(actually for technical reasons we also allow the
*
variable lớn be restricted further to lớn a subprogression of
*
}" class="latex" />, but let us ignore this minor extension for this discussion). There is some flexibility in how khổng lồ choose these approximants, but we eventually found it convenient to lớn use the following choices.

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For the von Mangoldt function
*
, we eventually went with the Cramér-Granville approximant
*
, where
*
, we used a somewhat complicated-looking approximant
*
for some explicit polynomials
*
, chosen so that
*
*
have almost exactly the same sums along arithmetic progressions (see the paper for details).

The objective is then to lớn obtain bounds on sums such as (1) that improve upon the “trivial bound” that one can get with the triangle inequality và standard number theory bounds such as the Brun-Titchmarsh inequality. For

*
*
, the Siegel-Walfisz theorem suggests that it is reasonable lớn expect error terms that have “strongly logarithmic savings” in the sense that they gain a factor of
*
over the trivial bound for any
*
0}" class="latex" />; for
*
, the Dirichlet hyperbola method suggests instead that one has “power savings” in that one should gain a factor of
*
over the trivial bound for some
*
0}" class="latex" />. In the case of the Möbius function
*
, there is an additional trick (introduced by Matomäki and Teräväinen) that allows one to lower the exponent
*
somewhat at the cost of only obtaining “weakly logarithmic savings” of shape
*
for some small
*
0}" class="latex" />.

Our main estimates on sums of the form (1) work in the following ranges:

For
*
, one can obtain weakly logarithmic savings for
*
. For
*
, one can obtain power savings for
*
. For
*
, one can obtain nguồn savings for
*
.

Conjecturally, one should be able khổng lồ obtain nguồn savings in all cases, & lower

*
down to lớn zero, but the ranges of exponents và savings given here seem to be the limit of current methods unless one assumes additional hypotheses, such as GRH. The
*
result for correlation against Fourier phases
*
was established previously by Zhan, & the
*
result for such phases và
*
was established previously by by Matomäki and Teräväinen.

By combining these results with tools from additive combinatorics, one can obtain a number of applications:

We now briefly discuss some of the ingredients of proof of our main results. The first step is standard, using combinatorial decompositions (based on the Heath-Brown identity & (for the

*
result) the Ramaré identity) to lớn decompose
*
into more tractable sums of the following types:

Type
*
sums
, which are basically of the form
*
for some weights
*
of controlled kích thước and some cutoff
*
that is not too large; Type
*
sums
, which are basically of the size
*
for some weights
*
,
*
of controlled kích thước and some cutoffs
*
that are not too close to
*
or to lớn
*
; Type
*
sums
, which are basically of the size
*
for some weights
*
of controlled kích thước and some cutoff
*
that is not too large.

The precise ranges of the cutoffs

*
depend on the choice of
*
; our methods fail once these cutoffs pass a certain threshold, và this is the reason for the exponents
*
being what they are in our main results.

The Type

*
sums involving nilsequences can be treated by methods similar to lớn those in this previous paper of Ben Green and myself; the main innovations are in the treatment of the Type
*
& Type
*
sums.

For the Type

*
sums, one can split into the “abelian” case in which (after some Fourier decomposition) the nilsequence
*
is basically of the khung
*
, & the “non-abelian” case in which
*
is non-abelian and
*
exhibits non-trivial oscillation in a central direction. In the abelian case we can adapt arguments of Matomaki & Shao, which uses Cauchy-Schwarz và the equidistribution properties of polynomials to obtain good bounds unless
*
is “major arc” in the sense that it resembles (or “pretends khổng lồ be”)
*
for some Dirichlet character
*
& some frequency
*
, but in this case one can use classical multiplicative methods khổng lồ control the correlation. It turns out that the non-abelian case can be treated similarly. After applying Cauchy-Schwarz, one ends up analyzing the equidistribution of the four-variable polynomial sequence

*

as
*
range in various dyadic intervals. Using the known multidimensional equidistribution theory of polynomial maps in nilmanifolds, one can eventually show in the non-abelian case that this sequence either has enough equidistribution to lớn give cancellation, or else the nilsequence involved can be replaced with one from a lower dimensional nilmanifold, in which case one can apply an induction hypothesis.

For the type

*
sum, a model sum to lớn study is

*
exponent for our
*
result can be as low as
*
.

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In a sequel lớn this paper (currently in preparation), we will obtain analogous results for almost all intervals

*
}" class="latex" /> with
*
in the range
*
}" class="latex" />, in which we will be able to lớn lower
*
all the way to lớn
*
.