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

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 $$\max_{x \in \mathbb{Z}_p} \mathbb{P}[Y = x] \leq C_{3,\lambda}\lambda.$$ 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.