Higher uniformity of arithmetic functions in short intervals I. All intervals
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866929258702569472 |
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| author | Matomäki, Kaisa Shao, Xuancheng Tao, Terence Teräväinen, Joni |
| author_facet | Matomäki, Kaisa Shao, Xuancheng Tao, Terence Teräväinen, Joni |
| contents | We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$.
More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ 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 $f,θ$. 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 multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_03754 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher uniformity of arithmetic functions in short intervals I. All intervals Matomäki, Kaisa Shao, Xuancheng Tao, Terence Teräväinen, Joni Number Theory 11N37, 11B30 We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$. More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ 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 $f,θ$. 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 multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums. |
| title | Higher uniformity of arithmetic functions in short intervals I. All intervals |
| topic | Number Theory 11N37, 11B30 |
| url | https://arxiv.org/abs/2204.03754 |