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.