The sharp form of the Kolmogorov--Rogozin inequality and a conjecture of Leader--Radcliffe

Let $X$ be a random variable and define its concentration function by $${\mathcal{Q}}_h(X)=\underset{x\in \mathbb{R}}{\sup }\mathbb{P}(X\in (x,x+h])\mathrm{.}$$ For a sum $S_n=X_1+\cdots +X_n$ of independent real-valued random variables the Kolmogorov-Rogozin inequality states that $${\mathcal{Q}}_h(S_n)\le C{(\sum _{i=1}^n(1-{\mathcal{Q}}_h(X_i)))}^{-\frac{1}{2}}\mathrm{.}$$ In this paper we give an optimal bound for ${\mathcal{Q}}_h(S_n)$ in terms of ${\mathcal{Q}}_h(X_i)$, which settles a question posed by Leader and Radcliffe in 1994. Moreover, we show that the extremal distributions are mixtures of two uniform distributions each lying on an arithmetic progression.