Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:

  1. $a_i-a_j\neq a_{i'}-a_{j'}$ holds if $i\neq i'$ or $j\neq j'$ or both.

  2. $\min_{1\leq i<j\leq m}|a_i-a_j|>b$ holds.

Denote $\mathcal T_m$ to be set of $b$-separated Sidon sets in $(p^\alpha,p-p^\alpha)$ with cardinality $m$.

Denote $\mathcal D(\mathcal R)$ to be set of differences $a_i-a_j$ represented by a $b$-separated Sidon set $\mathcal R\in\mathcal T_m$ (note $|\mathcal D(\mathcal R)|=\Omega(m^2)$).

Given $\beta\in[0,2)$ what is the maximum $n$ such that there are $\mathcal R_1,\dots,\mathcal R_n\in\mathcal T_m$ such that $|\mathcal D(\mathcal R_i)\cap\mathcal D(\mathcal R_j)|=O(m^{\beta})$ for all $1\leq i<j\leq n$?


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