Let $\epsilon <1/2$. Let $X$ be a random variable in $\mathbb Z$ such that $\mathbb P (X=x)\le \epsilon $ for any $x\in \mathbb Z$ (you may add any moment or regularity conditions on $X$ if needed). Let $S\_n$ be a sum of $n$ independent copies of $X$. Show that for any $x\in \mathbb Z$
$$\mathbb P (S\_n=x) \le C\cdot \epsilon \cdot n^{-1/2}$$
for some universal constant $C$.


I'm also looking for high dimensional generalizations of this claim in which the factor $n^{-1/2}$ is replaced with $n^{-d/2}$ where $d$ is the dimension (I'm thinking of $d$ as fixed). Clearly, for such a statement to hold we need an assumption saying that $X$ is "truly $d$ dimensional". I wonder if the following statement is correct:


Let $M$ large and let $X$ be a random variable in $\mathbb Z ^d$ such that $\mathbb P (X=x)\le M^{-d/2}$ for any $x\in \mathbb Z ^d$ and $\mathbb P(X\cdot v=y) \le 1/M$ for any vector $v\in \mathbb R ^d$ and $y>0$. Then, for any $x\in \mathbb Z ^d$ we have
$$\mathbb P (S\_n=x) \le C\_d (Mn)^{-d/2}$$
for some constant $C\_d$ depending only on $d$.



Local probabilities for lattice random walk
For the one dimensional case, a quite nice bound is in Theorem 4.2 of [1]. See also [2]. The dependence on $\epsilon$ that you seek was first shown by Kesten[3].


The combinatorial approach was revived in [4]. The sharpest result is quite recent, see [5], which also identifies the worst case. Look there first.


[1] Esseen, Carl-Gustav. "On the concentration function of a sum of independent random variables." Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 9, no. 4 (1968): 290-308. <https://link.springer.com/content/pdf/10.1007/BF00531753.pdf>


[2] <https://link.springer.com/article/10.1023/A:1022654631571>


[3] KESTEN, H. (1969). A sharper form of the Doeblin-Lévy-Kolmogorov-Rogozin inequality for concentration functions. Math. Scand. 25 133–144.


[4] LEADER, I. and RADCLIFFE, A. J. (1994). Littlewood-Offord inequalities for random variables. SIAM J.
Discrete Math. 7 90–101.


[5] <https://arxiv.org/pdf/2201.09861.pdf>


