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arXiv · 2609.08724

On the cardinality of the exception set in Littlewood's discrete conjecture

A. A. Illarionov
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Abstract

Let $p$ be a prime, $d\ge 2$, $H\in [1,p)$, and $\ln\ln p = o(\ln H)$. We prove that $$ \min_{1\le n \le H} n \left\|\frac{a_1 n}{p}\right\|\ldots \left\|\frac{a_d n}{p}\right\| \approx \frac{1}{(\ln p)^{d-1} \ln H} $$ for "almost all" $a \in ({\Bbb Z} / p{\Bbb Z})^d$.

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A. A. Illarionov. 2026-09-08. On the cardinality of the exception set in Littlewood's discrete conjecture. https://arxiv.org/abs/2609.08724

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