Anticoncentration of Random Sums in ${\mathbb{Z}}_p$

Abstract

In this paper we investigate the probability distribution of the sum $Y$ of $\ell$ independent identically distributed random variables taking values in ${\mathbb{Z}}_p$. Our main focus is the regime of small values of $\ell$, which is less explored compared to the asymptotic case $\ell \to \infty$. Starting with the case $\ell =3$, we prove that if the distributions of the $Y_i$ are uniformly bounded by $\lambda <1$ and $p>2/\lambda$, then there exists a constant $C_{3,\lambda }<1$ such that \[ \max_{x \in \mathbb{Z}_p} \mathbb{P}[Y = x] \leq C_{3,λ}λ. \] Moreover, when the distributions are uniformly separated from $1$, the constant $C_{3,\lambda }$ can be made explicit. By iterating this argument, we obtain effective anticoncentration bounds for larger values of $\ell$, yielding nontrivial estimates already in small and moderate regimes where asymptotic results do not apply.

arXiv commentThis manuscript provides a substantial revision and a significant refocusing of the earlier preprint arXiv:2308.04284. The current version removes the applications to sequenceability to prioritize the development of anticoncentration inequalities in ${\mathbb{Z}}_p$ and provides a more detailed comparison with the existing literature