# Number of $b$-separated Sidon sets with pairwise difference set intersection bounds

Given two integers integers $$0 and a real $$\alpha\in(0,1)$$ call a set of $$m$$ integers $$a_1<\dots 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. $$\min_{1\leq ib$$ 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?