I was interested in the following question, which I am not sure if it is true or not. I would appreciate any answers or references. Thank you! (Please note this is the revised version of the original question asked as pointed out by the comments of Mr/Ms The Masked Avenger)

Suppose I have real numbers $0 < \theta < \gamma < 1$. Does there exist some absolute constant $C> 0$ such that the following holds for $X$ sufficiently large?: If $X > q > X^{\gamma}$, then we can find $ X^{\theta}/ \log X < k < X^{\theta}\log X$ such that $q \equiv r (\text{mod }k)$ and $0 \leq r \leq C \log X$. (or possibly the same statement where we replace $\log X$ with $X^{\epsilon}$)

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    $\begingroup$ No. Let K be the product of all the candidate k in the range specified when X and theta are given. For your favorite integer A let q= AK + C + 1. Then q will be = C+1 mod k for any of your desired k. When X gets big enough, your constant C does not work. $\endgroup$ – The Masked Avenger May 8 '14 at 22:01
  • $\begingroup$ Even if you further restricted q , say q at most X, I would be surprised to find a result. I can imagine C depending on logX though. $\endgroup$ – The Masked Avenger May 8 '14 at 22:06
  • $\begingroup$ Thank you for your comments. I greatly appreciate it! I think I am going to change the question accordingly. $\endgroup$ – SJY May 8 '14 at 22:08
  • $\begingroup$ As you wish. I would appreciate the revised version giving a nod to the comments, so that what I said earlier is not totally out of place. $\endgroup$ – The Masked Avenger May 8 '14 at 22:12
  • $\begingroup$ Hope this works! Thank you for your helpful comments! $\endgroup$ – SJY May 8 '14 at 22:18

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