**Edit:** This is a question related to my other post, stated in a much more concrete way I think.

I am interested in anything (ideas, references) related to the following problem:

Suppose that $A \subset \mathbb{Z}_p$ is a set of size $\delta p$ for $\delta > \frac{1}{2}$ (i.e. relatively big). What is known about the distribution of $\mu_A*\mu_A$ compared to $\mu_A$? In particular, what is known about $$\left\|\mu_A - \mu_A*\mu_A \right\|_{\ell_1(G)}$$

Taking $A = \left(\frac{p}{3},\frac{2p}{3}\right)$ we obtain that $A+A$ could be almost completely disjoint with $A$. This is however only possible if $\delta < \frac{1}{3}$ in view of the Cauchy-Davenport Theorem.

But even for $\delta>\frac{1}{3}$, it might still happen that $\mu_A*\mu_A$ puts a lot of its mass outside of $A$ and the distance is large even if $A+A$ intersects $A$. Do you know any concrete quantitative bounds?

**Motivation:** The upper bound would give us the bound on the rate of convergence of $n$-fold convolution of every measure on $\mathbb{Z}_p$, that has is $\delta$-close to the uniform measure in the total variational distance.