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arXiv · 1812.01224

Fourier uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki·Maksym Radziwiłł·Terence Tao
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Abstract

Let $λ$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_α \left | \sum_{x < n \leq x + H} λ(n) e(-αn) \right | dx = o ( X H) $$ for all $H \geq X^θ$ with $θ> 0$ fixed but arbitrarily small. Previously, this was only known for $θ> 5/8$. For smaller values of $θ$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $λ(n) Λ(n + h) Λ(n + 2h)$ over the ranges $h < X^θ$ and $n < X$, and where $Λ$ is the von Mangoldt function.

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Kaisa Matomäki, Maksym Radziwiłł, Terence Tao. 2018-12-04. Fourier uniformity of bounded multiplicative functions in short intervals on average. https://arxiv.org/abs/1812.01224

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