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

Journal article

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.

Authors: Matomäki, K; Shao, X; Tao, T; Teräväinen, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press (CUP)
• Journal: Forum of Mathematics Pi
• Volume: 11
• Pages: e29
• Article number: e29
• Publication date: 2023-10-19
• DOI: 10.1017/fmp.2023.28
• ISSN: 2050-5086
• Language: English
• Keywords: Ergodic theory; Geometry; Mathematical analysis; Mathematics; Arithmetic; Combinatorics; Type (biology); Function (biology); Physics; Interval (graph theory); Lambda; Divisor (algebraic geometry); Hyperbola; Quantum mechanics