Problem 67

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that

\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved: Tao's 2015 theorem, refereed in Discrete Analysis in 2016, answers yes; see the claim page (Tao, 2015).

Er64b: Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.
Er65b: Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
Er81: Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
Er85c: Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.
Mc21: McNamara, Redmond, Dynamical methods for the Sarnak and Chowla conjectures. PhD dissertation, University of California, Los Angeles (2021), Chapter 4; https://escholarship.org/uc/item/4wr015m0.
Ta16: Tao, Terence, The Erdős discrepancy problem. Discrete Anal. (2016), Paper No. 1, 27 pp.