Problem 992

Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$ , the discrepancy

D(N)=\max_{I\subseteq [0,1]} \lvert \#\{ n\leq N : \{ \alpha x_n\}\in I\} - \lvert I\rvert N\rvert

satisfies

D(N) \ll N^{1/2}(\log N)^{o(1)}?

Or even

D(N)\ll N^{1/2}(\log\log N)^{O(1)}?

Source: erdosproblems.com/992

Discrepancy

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved: Berkes and Philipp [BePh94] built an integer sequence whose discrepancy has $\limsup D(N)/(N\log N)^{1/2}>0$ for almost every $\alpha$ , so both asked bounds fail; the accepted claim is Berkes and Philipp 1994.

Ba81: Baker, R. C., Metric number theory and the large sieve. J. London Math. Soc. (2) (1981), 34-40.
BePh94: Berkes, István and Philipp, Walter, The size of trigonometric and Walsh series and uniform distribution ${\rm mod}\ 1$ . J. London Math. Soc. (2) (1994), 454-464.
Ca50: Cassels, J. W. S., Some metrical theorems of Diophantine approximation. III. Proc. Cambridge Philos. Soc. (1950), 219-225.
ErKo49: Erdős, P. and Koksma, J. F., On the uniform distribution modulo $1$ of sequences $(f(n,\theta))$ . Nederl. Akad. Wetensch., Proc. (1949), 851-854 = Indagationes Math. 11, 299-302.

Problem 992

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