Saturday, October 1, 2022
HomeMathIncreased uniformity of arithmetic capabilities briefly intervals I. All intervals

# Increased uniformity of arithmetic capabilities briefly intervals I. All intervals

Kaisa Matomäki, Xuancheng Shao, Joni Teräväinen, and myself have simply uploaded to the arXiv our preprint “Increased uniformity of arithmetic capabilities briefly intervals I. All intervals“. This paper investigates the upper order (Gowers) uniformity of normal arithmetic capabilities in analytic quantity idea (and particularly, the Möbius perform ${mu}$, the von Mangoldt perform ${Lambda}$, and the generalised divisor capabilities ${d_k}$) briefly intervals ${(X,X+H]}$, the place ${X}$ is massive and ${H}$ lies within the vary ${X^{theta+varepsilon} leq H leq X^{1-varepsilon}}$ for a hard and fast fixed ${0 < theta < 1}$ (that one wish to be as small as potential). If we let ${f}$ denote one of many capabilities ${mu, Lambda, d_k}$, then there’s intensive literature on the estimation of brief sums

$displaystyle sum_{X < n leq X+H} f(n)$

and a few literature additionally on the estimation of exponential sums akin to

$displaystyle sum_{X < n leq X+H} f(n) e(-alpha n)$

for an actual frequency ${alpha}$, the place ${e(theta) := e^{2pi i theta}}$. For purposes within the additive combinatorics of such capabilities ${f}$, it is usually mandatory to contemplate extra basic correlations, akin to polynomial correlations

$displaystyle sum_{X < n leq X+H} f(n) e(-P(n))$

the place ${P: {bf Z} rightarrow {bf R}}$ is a polynomial of some fastened diploma, or extra usually

$displaystyle sum_{X < n leq X+H} f(n) overline{F}(g(n) Gamma)$

the place ${G/Gamma}$ is a nilmanifold of fastened diploma and dimension (and with some management on construction constants), ${g: {bf Z} rightarrow G}$ is a polynomial map, and ${F: G/Gamma rightarrow {bf C}}$ is a Lipschitz perform (with some certain on the Lipschitz fixed). Certainly, due to the inverse theorem for the Gowers uniformity norm, such correlations let one management the Gowers uniformity norm of ${f}$ (probably after subtracting off some renormalising issue) on such brief intervals ${(X,X+H]}$, which may in flip be used to manage different multilinear correlations involving such capabilities.

Historically, asymptotics for such sums are expressed by way of a “most important time period” of some arithmetic nature, plus an error time period that’s estimated in magnitude. For example, a sum akin to ${sum_{X < n leq X+H} Lambda(n) e(-alpha n)}$ can be approximated by way of a most important time period that vanished (or is negligible) if ${alpha}$ is “minor arc”, however can be expressible by way of one thing like a Ramanujan sum if ${alpha}$ was “main arc”, along with an error time period. We discovered it handy to cancel off such most important phrases by subtracting an approximant ${f^sharp}$ from every of the arithmetic capabilities ${f}$ after which getting higher bounds on the rest correlations akin to

$displaystyle |sum_{X < n leq X+H} (f(n)-f^sharp(n)) overline{F}(g(n) Gamma)| (1)$

(truly for technical causes we additionally enable the ${n}$ variable to be restricted additional to a subprogression of ${(X,X+H]}$, however allow us to ignore this minor extension for this dialogue). There’s some flexibility in how to decide on these approximants, however we finally discovered it handy to make use of the next decisions.

The target is then to acquire bounds on sums akin to (1) that enhance upon the “trivial certain” that one can get with the triangle inequality and customary quantity idea bounds such because the Brun-Titchmarsh inequality. For ${mu}$ and ${Lambda}$, the Siegel-Walfisz theorem means that it’s cheap to count on error phrases which have “strongly logarithmic financial savings” within the sense that they acquire an element of ${O_A(log^{-A} X)}$ over the trivial certain for any ${A>0}$; for ${d_k}$, the Dirichlet hyperbola methodology suggests as an alternative that one has “energy financial savings” in that one ought to acquire an element of ${X^{-c_k}}$ over the trivial certain for some ${c_k>0}$. Within the case of the Möbius perform ${mu}$, there’s an extra trick (launched by Matomäki and Teräväinen) that enables one to decrease the exponent ${theta}$ considerably at the price of solely acquiring “weakly logarithmic financial savings” of form ${log^{-c} X}$ for some small ${c>0}$.

Our most important estimates on sums of the shape (1) work within the following ranges:

Conjecturally, one ought to have the ability to acquire energy financial savings in all instances, and decrease ${theta}$ all the way down to zero, however the ranges of exponents and financial savings given right here appear to be the restrict of present strategies until one assumes further hypotheses, akin to GRH. The ${theta=5/8}$ end result for correlation towards Fourier phases ${e(alpha n)}$ was established beforehand by Zhan, and the ${theta=3/5}$ end result for such phases and ${f=mu}$ was established beforehand by by Matomäki and Teräväinen.

By combining these outcomes with instruments from additive combinatorics, one can acquire various purposes:

• Direct insertion of our bounds within the current work of Kanigowski, Lemanczyk, and Radziwill on the prime quantity theorem on dynamical programs which might be analytic skew merchandise provides some enhancements within the exponents there.
• We are able to acquire a “brief interval” model of a a number of ergodic theorem alongside primes established by Frantzikinakis-Host-Kra and Wooley-Ziegler, through which we common over intervals of the shape ${(X,X+H]}$ relatively than ${[1,X]}$.
• We are able to acquire a “brief interval” model of the “linear equations in primes” asymptotics obtained by Ben Inexperienced, Tamar Ziegler, and myself in this sequence of papers, the place the variables in these equations lie briefly intervals ${(X,X+H]}$ relatively than lengthy intervals akin to ${[1,X]}$.

We now briefly talk about among the substances of proof of our most important outcomes. Step one is customary, utilizing combinatorial decompositions (primarily based on the Heath-Brown id and (for the ${theta=3/5}$ end result) the Ramaré id) to decompose ${mu(n), Lambda(n), d_k(n)}$ into extra tractable sums of the next sorts:

The exact ranges of the cutoffs ${A, A_-, A_+}$ rely upon the selection of ${theta}$; our strategies fail as soon as these cutoffs cross a sure threshold, and that is the explanation for the exponents ${theta}$ being what they’re in our most important outcomes.

The Kind ${I}$ sums involving nilsequences could be handled by strategies just like these in this earlier paper of Ben Inexperienced and myself; the principle improvements are within the remedy of the Kind ${II}$ and Kind ${I_2}$ sums.

For the Kind ${II}$ sums, one can cut up into the “abelian” case through which (after some Fourier decomposition) the nilsequence ${F(g(n)Gamma)}$ is mainly of the shape ${e(P(n))}$, and the “non-abelian” case through which ${G}$ is non-abelian and ${F}$ reveals non-trivial oscillation in a central course. Within the abelian case we will adapt arguments of Matomaki and Shao, which makes use of Cauchy-Schwarz and the equidistribution properties of polynomials to acquire good bounds until ${e(P(n))}$ is “main arc” within the sense that it resembles (or “pretends to be”) ${chi(n) n^{it}}$ for some Dirichlet character ${chi}$ and a few frequency ${t}$, however on this case one can use classical multiplicative strategies to manage the correlation. It seems that the non-abelian case could be handled equally. After making use of Cauchy-Schwarz, one finally ends up analyzing the equidistribution of the four-variable polynomial sequence

$displaystyle (n,m,n',m') mapsto (g(nm)Gamma, g(n'm)Gamma, g(nm') Gamma, g(n'm'Gamma))$

as ${n,m,n',m'}$ vary in varied dyadic intervals. Utilizing the recognized multidimensional equidistribution idea of polynomial maps in nilmanifolds, one can finally present within the non-abelian case that this sequence both has sufficient equidistribution to provide cancellation, or else the nilsequence concerned could be changed with one from a decrease dimensional nilmanifold, through which case one can apply an induction speculation.

For the kind ${I_2}$ sum, a mannequin sum to review is

$displaystyle sum_{X < n leq X+H} d_2(n) e(alpha n)$

which one can develop as

$displaystyle sum_{n,m: X < nm leq X+H} e(alpha nm).$

We experimented with various methods to deal with the sort of sum (together with automorphic kind strategies, or strategies primarily based on the Voronoi components or van der Corput’s inequality), however considerably to our shock, probably the most environment friendly strategy was an elementary one, through which one makes use of the Dirichlet approximation theorem to decompose the hyperbolic area ${{ (n,m) in {bf N}^2: X < nm leq X+H }}$ into various arithmetic progressions, after which makes use of equidistribution idea to ascertain cancellation of sequences akin to ${e(alpha nm)}$ on nearly all of these progressions. Because it seems, this technique works nicely within the regime ${H > X^{1/3+varepsilon}}$ until the nilsequence concerned is “main arc”, however the latter case is treatable by current strategies as mentioned beforehand; this is the reason the ${theta}$ exponent for our ${d_2}$ end result could be as little as ${1/3}$.

In a sequel to this paper (presently in preparation), we are going to acquire analogous outcomes for virtually all intervals ${(x,x+H]}$ with ${x}$ within the vary ${[X,2X]}$, through which we can decrease ${theta}$ all the best way to ${0}$.

RELATED ARTICLES