Let $(X\_i)\_{i\in \mathbb{N}}$ iid random variables such that there exists $\alpha>0$ where $\mathbb{P}\left(X\_1\in [x,x+1]\right)\leq \alpha$ for all $x\in \mathbb{R}$. Assume $\alpha$ small enough, does there exist a universal constant $C>0$ so that $$\mathbb{P}\left(\sum\_{i=1}^N X\_i\in [x,x+1]\right)\leq \frac{C\alpha}{\sqrt{N}}$$ for all $x\in\mathbb{R}$ and $N\in \mathbb{N}^\*$?


A few remarks :


* From the central limit theorem this is very natural to be expected. However here we do not assume $\mathbb{E}(X^2)<\infty$ and it is not so clear to me how to use the characteristic functions with the hypothesis.
* In the case of discrete variable $X\in \mathbb{Z}$, one should just do the computation. However I can't see why should it be the "worse case scenario" and how to compare it with the more general case.



Regularity for the sum of iid random variables
Yes, this is a special case of an inequality by Kesten (see [Theorem 2 and Corollary 1](https://www.mscand.dk/article/view/10950/8971)).


In particular, letting $L=\lambda=:t$ in that Corollary 1, we get the following:



> 
> Let $X\_1,\dots,X\_n$ be independent identically distributed random variables, with $S\_n:=\sum\_1^n X\_k$. Then for any positive real $t$,
> $$Q(S\_n,t)\le\frac{CQ(X\_1,t)}{\sqrt{(1-Q(X\_1,t))\, n}},$$
> where $C$ is a universal positive real constant and $Q(X,u):=\sup\_{x\in\mathbb R}P(x\le X\le x+u)$ for positive real $u$.
> 
> 
> 


