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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!

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When is something "very well distributed"? I have never heard the term before. –  Sanath May 12 at 21:58
    
oops. let me fix that. –  SJY May 12 at 22:01
    
Have a look at mathoverflow.net/questions/163407/…. I think my answer gives an example –  Anthony Quas May 12 at 22:51
    
Do you have access to the Kuipers-Niederreiter book, SJY? That's the first place I'd look. –  Gerry Myerson May 12 at 23:54
    
@AnthonyQuas Thank you for your comment! Does the "Ergodic Theorem" you are using in your solution, give a bound on the error by any chance? I can't quite see how this is an example yet. It would be nice if it is. –  SJY May 12 at 23:59

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