I have been thinking about some set that is equidistributed modulo $q$, uniformly in $q$ in some sense. I was starting to think this particular condition, which I describe below, is too strong and that I am starting to doubt if any such set even exists.

However, I haven't been able to prove no such set exists. I was wondering if someone could tell me how to prove that such set does not exist (or that does exist). I would also appreciate any input or discussion on whether such set exists or not. It would be great if I could get a better understanding! Thank you very much!

Let $\gamma >\theta >0$, and some $\epsilon >0$ small. Let $B \subseteq \mathbb{N}$ and $B \cap [1,X] \sim X^{\gamma}$. Suppose given any $q \in \mathbb{N}$ and any partition of the residue class modulo $q$, $\{ J_l \}_{l=1}^L$, such that $|J_l|X^{\gamma}/q > X^{\epsilon}$, we have $$ \sum_{l=1}^L| \ \# \{ u \in B : u \leq X, u \equiv J_l (\text{mod }q) \} - (|J_l|X^{\gamma})/q \ | \ll X^{\gamma - \theta}. $$ I am wondering does such a set $B$ exist? If $J = \{ r_1, .., r_k \}$, by $u \equiv J (\text{mod }q)$, I mean $u \equiv r_j (\text{mod }q)$ for some $j \in \{1, ..., k \}$. Thanks!