Anticoncentration of Random Sums in $\mathbb{Z}_p$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910026085433344 |
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| author | Costa, Simone |
| author_facet | Costa, Simone |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anticoncentration of Random Sums in $\mathbb{Z}_p$ Costa, Simone Probability Combinatorics Number Theory 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. |
| title | Anticoncentration of Random Sums in $\mathbb{Z}_p$ |
| topic | Probability Combinatorics Number Theory |
| url | https://arxiv.org/abs/2602.16595 |