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.
The Littlewood-Offord problem is a classical question in probabilistic combinatorics concerning the anticoncentration of sums of independent random variables. It was introduced by Littlewood and Offord in the 1940s (see [18]) in order to study the distribution of sums of random variables with restricted supports. In its original formulation, given a list of (not necessarily distinct) integers, the problem asks for upper bounds on the probability of obtaining a prescribed value by summing elements from the list. Equivalently, given integers (v 1 , v 2 , . . . , v ℓ ), one studies the distribution of ℓ i=1 Y i , where each Y i is uniformly distributed on {0, v i } (or, in an equivalent formulation, on {-v i , v i }). In [18], Littlewood and Offord proved an upper bound of order O( log n √ n ), which was later improved by Erdős to 1 √ n (1 + o(1)) in [7]. The problem and its variants arise in the analysis of random walks, random matrices, and other combinatorial structures (see for instance [4,24]). Significant progress has been made under various structural assumptions on the distributions of the variables (see [13,14]), as well as in settings involving finite groups. In particular, Vaughan and Wooley [25] considered the case where the variables Y i are uniform on {0, v i } in cyclic groups, and further bounds were obtained by Griggs [9], Bibak [4], and Juskevicius and Semetulskis [12]. In the latter work, the authors also investigated the case where the distributions of the Y i are uniformly bounded by 1/2.
The present work is inspired by these developments. We consider independent random variables Y 1 , . . . , Y ℓ taking values in a cyclic group Z k , whose distributions are pointwise bounded by λ.
If the distributions are uniform on a subset A ⊆ Z k with |A| = n, the problem is closely related to estimating the probability that a subset X ⊆ A of fixed cardinality ℓ has sum equal to x. This can be viewed as a variation of the classical Littlewood-Offord problem in which the size of the subset is fixed and the elements v 1 , . . . , v n are distinct. The case of distinct integers was studied by Erdős and later by Halász in [10] (see also [23]), where bounds of order O( 1 n √ n ) were obtained; very recent results have been obtained for this problem also in Z p by Pham and Sauermann (see [21]). These questions are also motivated by applications to the set sequenceability problem (see [5,11] and [21]), where one needs anticoncentration estimates in finite cyclic groups in regimes where both p and ℓ are sufficiently large. In this context, it was brought to our attention that Lev established very strong asymptotic results in [15,16]. In particular, in [16] he proved that in Z p one has
Thus, for p and ℓ sufficiently large, one recovers the same qualitative behaviour as in the integer setting. However, a direct inspection shows that for ℓ < 24 the above estimate does not improve upon the trivial bound. The aim of this paper is to address precisely this non-asymptotic regime. We will moreover treat distributions that are not necessarily uniform.
The paper is organized as follows. In Section 2 we revisit the integer case. We present a direct and self-contained proof of a bound that appears in [13] in a more general framework, and we make the constants explicit, improving in particular the case ℓ = 3. Denoting by n the cardinality of the support, we show that there exists an absolute constant D such that
In particular, for every ϵ > 0, if ℓ is sufficiently large with respect to ϵ, then
Moreover, in the case ℓ = 3, we obtain the explicit bound ϵ = 3+1/n 2
, which is already non-trivial. We then turn to cyclic groups. When the prime factors of k are sufficiently large, a Freiman isomorphism of order ℓ allows us to transfer the integer bound to Z k . To avoid assumptions on the prime factors of k, one may alternatively use Lev’s results.
Our main contribution is contained in Section 3, where we obtain non-trivial anticoncentration bounds for every ℓ ≥ 3. More precisely, if p > 2 λ ℓ 0 3 ν (where ℓ 0 is a power of three such that ℓ 0 ≤ ℓ), and if
. and distributions uniformly bounded by λ ≤ 9/10 (this restriction is only needed to provide an explicit value of ν), then there exists an absolute constant ν > 0 such that
In summary, while Lev’s asymptotic methods dominate in the large-ℓ regime, our approach provides explicit and self-contained bounds for small and moderate values of ℓ, and applies also to distributions that are not necessarily uniform, where asymptotic estimates are not yet effective.
In this section, given a set A = {v 1 , v 2 , . . . , v n } of distinct integers, we consider the random variable Y given by the sum of ℓ independent variables Y 1 , . . . , Y ℓ that are uniformly distributed on A. We establish an upper bound on P[Y = x] corresponding to a special case of Theorem 2.3 in [13]. Although this follows from their more general result, we provide a direct and self-contained proof, which
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