Affiliation:
1. Department of Pure Mathematics, Tel Aviv University , Haim Levanon Street 30, Tel Aviv 6997801, Israel
Abstract
Abstract
We study the minimal gap statistic for sequences of the form $\left( \alpha x_n \right)_{n = 1}^{\infty}$ where $\left( x_n \right)_{n = 1}^{\infty}$ is a sequence of real numbers, and its connection to the additive energy of $\left( x_n \right)_{n = 1}^{\infty}$. Inspired by a recent paper of Aistleitner, El-Baz and Munsch we show conditionally on the Lindelöf Hypothesis that if the additive energy is of lowest possible order then for almost all α, the minimal gap $\delta_{\min}^{\alpha} (N) = \min \left\{\alpha x_m - \alpha x_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$ is close to that of a random sequence, a result Rudnick showed for integer-valued sequences. We also show unconditional results in this direction, as well as some converse theorems about sequences with large additive energy.
Funder
European Union‘s Horizon 2020 research
Publisher
Oxford University Press (OUP)
Cited by
2 articles.
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