Problem 67

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claims/: The 1 claim page of Problem 67, one per claimant's result; the problem's standing derives from them.

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

Source. erdosproblems.com/67, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #67, https://www.erdosproblems.com/67.

References.

• [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.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1979_old_new_problems_results_combinatorial_number
• erdos_1964_problems_results_diophantine_approximations
• tao_2016_erdos_discrepancy_problem
• erdos_1962_szamelmeleti_megjegyzesek_iv
• erdos_1965_recent_advances_current_problems_number_theory
• erdos_1981_combinatorial_problems_which_i_would_most