OP-00957 Open

A dilate with no short interval holding many points

Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Konyagin, Ruzsa and Schlag proved the statement below for $c > \frac13$.

Problem. Is there a $\theta$ for which every interval of length $\frac1n$ in $\mathbb{R}/\mathbb{Z}$ contains at most $n^c$ of the points $\theta a \bmod 1$, $a \in A$?

Source: J. Komlós and I. Z. Ruzsa; H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis (1994), Problem 17(4). Listed as Problem 86 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: NT — Number Theory
Secondary: CA, CO
Keywords: dilates, discrepancy, uniform distribution modulo one
Literature checked: Oct. 26, 2026 by clemens-admin

References

@book{montgomery1994,
  author = {Montgomery, Hugh L.},
  title = {Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis},
  series = {CBMS Regional Conference Series in Mathematics},
  volume = {84},
  publisher = {American Mathematical Society},
  year = {1994},
  doi = {10.1090/cbms/084}
}

@article{konyagin2000,
  author = {Konyagin, S. V. and Ruzsa, I. Z. and Schlag, W.},
  title = {On uniformly distributed dilates of finite integer sequences},
  journal = {Journal of Number Theory},
  volume = {82},
  number = {2},
  pages = {165--187},
  year = {2000},
  doi = {10.1006/jnth.1999.2420}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 86; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

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OP-00957, "A dilate with no short interval holding many points", clopen.solutions.