Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract

We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

Bracket words along Hardy field sequences

“…However, it has been resolved in a number of special cases [Bou13, BSZ13, DDM15, DK15, eAKL16, eALdlR14, FKPLM16, GT12a, Gre12, KPL15, LS15, MR10, MR15, Mül17, Pec18, Vee16]; see also the recent survey articles [DLMR,FKPL18]. Of particular importance to the current paper is Möbius orthogonality for nilsequences [GT12a], which was recently strengthened to short intervals [MSTT22]. As we discuss later in this paper, this is closely connected to bracket words thanks to the work of Bergelson and Leibman [BL07].…”

“…We are then left with the task of establishing cancellation in each of these intervals. A key ingredient is Möbius orthogonality for nilsequences in short intervals, Theorem 5.3, recently established in [MSTT22]. The main technical difficulty of our argument lies in extending Theorem 5.3 to piecewise constant (and hence necessarily not continuous) functions with semialgebraic pieces, which we accomplish in §5.2.…”