Littlewood-Offord bounds on the symmetric groups and applications

Abstract: The anti-concentration phenomenon in probability theory has been intensively studied in recent years, with applications across many areas of mathematics. In most existing works, the ambient probability space is a product space generated by independent random variables. In this paper, we initiate a systematic study of anti-concentration when the ambient space is the symmetric group, equipped with the uniform measure. Concretely, we focus on the random sum $S_π = \sum_{i=1}^{n} w_i\, v_{π(i)}$, where $w=(w_1,\dots,w_n)$ and $v=(v_1,\dots,v_n)$ are fixed vectors and $π$ is a uniformly random permutation. The paper contains several new results, addressing both discrete and continuous anti-concentration phenomena. On the discrete side, we establish a near-optimal structural characterization of the vectors $w$ and $v$ under the assumption that the concentration probability $\sup_x P(S_π=x)$ is polynomially large. As applications, we derive and strengthen a number of previous results. In particular, we show that if both $w$ and $v$ have distinct entries, then $\sup_x P(S_π=x) \le n<sup>{-5/2+o(1)}$. This bound serves as a permutation-space analogue of the classical Erdős--Moser bound in the product-space setting and answers a question posed by Alon--Pohoata--Zhu. From the continuous perspective, we study the small-ball event $|S_π-L|\le δ$. We establish sharp bounds in various settings, including results exhibiting sub-gaussian decay in $L$. With additional effort, we are also able to treat the joint distribution of these events. Moreover, we provide a characterization of the vectors $w$ and $v$ for which these small-ball probabilities are large. As an application, we prove that the number of extremal points of random permutation polynomials is bounded by $O(\log n)$, extending results of S{ö}ze on the number of real roots.