On Littlewood-Offord theory for arbitrary distributions

Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When $\mathbb{P}(X_i=\pm 1) = \frac {1} {2}$, this corresponds to the classical Littlewood-Offord problem. We prove that in general for identically distributed random vectors and even values of $n$ the optimal choice for $(a_i)$ is $a_i=1$ for $i\leq \frac{n}{2}$ and $a_i=-1$ for $i > \frac {n} 2$, regardless of the distribution of $X_1$. Applying these results for Bernoulli random variables answers a recent question of Fox, Kwan and Sauermann. Finally, we provide sharp bounds for concentration probabilities of sums of random vectors under the condition $\sup_{x}\mathbb{P}(X_i=x)\leq \alpha$, where it turns out that the worst case scenario is provided by distributions on an arithmetic progression that are in some sense as close to the uniform distribution as possible. An important feature of this work is that unlike much of the literature on the subject we use neither methods of harmonic analysis nor those from extremal combinatorics.