Anticoncentration of Random Sums in $\mathbb{Z}_p$
Simone Costa
TL;DR
This work studies anticoncentration for sums $Y=Y_1+\dots+Y_{\ell}$ of i.i.d. variables in the finite group $\mathbb{Z}_p$, with a focus on small to moderate $\ell$ where asymptotic results fail. It provides a self-contained bound for the integer case, $\max_{x} \mathbb{P}[Y=x] \le \frac{D}{n\sqrt{\ell-1}}$, and a sharp $\ell=3$ bound $\max_{x} \mathbb{P}[Y=x] \le \frac{3+1/n^2}{4n}$, then extends these insights to $\mathbb{Z}_k$ via Freiman isomorphisms when prime factors are large, or via Lev-type bounds in the $\mathbb{Z}_p$ setting. For all $\ell\ge 3$, the paper proves nontrivial anticoncentration bounds under explicit conditions, notably $p > \frac{2}{\lambda}(\ell_0/3)^{\nu}$ with $\ell_0\le \ell$ a power of three and $\lambda \le 9/10$, yielding $\max_{x} \mathbb{P}[Y=x] \le \lambda(3/\ell_0)^{\nu}$ for some absolute $\nu>0$. Together, these results bridge non-asymptotic small/moderate-$\ell$ behavior with known large-$\ell$ asymptotics, providing computable anticoncentration bounds relevant for random sums in finite cyclic groups.
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 $λ< 1$ and $p > 2/λ$, then there exists a constant $C_{3,λ} < 1$ 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 $C_{3,λ}$ 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.