An asymptotically optimal bound for the concentration function of a sum of independent integer random variables

For a random variable $X$ define $Q(X)={\sup }_{x\in \mathbb{R}}\mathbb{P}(X=x)$. Let $X_1,\dots ,X_n$ be independent integer random variables. Suppose $Q(X_i)\le {\alpha }_i\in (0,1]$ for each $i\in \{1,\dots ,n\}$. Juškevičius (2023) conjectured that $Q(X_1+\cdots +X_n)\le Q(Y_1+\cdots +Y_n)$ where $Y_1,\dots ,Y_n$ are independent and $Y_i$ is a random integer variable with $Q(Y_i)={\alpha }_i$ that has the smallest variance, i.e. the distribution of $Y_i$ has probabilities ${\alpha }_i,\dots ,{\alpha }_i,{\beta }_i$ or probabilities ${\beta }_i,{\alpha }_i,\dots ,{\alpha }_i$ on some interval of integers, where $0\le {\beta }_i<{\alpha }_i$. We prove this conjecture asymptotically: i.e., we show that for each $\delta >0$ there is $V_0=V_0(\delta )$ such that if $\mathrm{V}ar(\sum Y_i)\ge V_0$ then $Q(\sum X_i)\le (1+\delta )Q(\sum Y_i)$. This implies an analogous asymptotically optimal inequality for concentration at a point when $X_1$, $\dots$, $X_n$ take values in a separable Hilbert space. Our long and technical argument relies on several non-trivial previous results including an inverse Littlewood–Offord theorem and an approximation in total variation distance of sums of multivariate lattice random vectors by a discretized Gaussian distribution.