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Claim. For every $f:\mathbb{N}\to\{-1,+1\}$ and every $C>0$ there are $d,m\ge1$ with

$$\left\lvert\sum_{1\le k\le m} f(kd)\right\rvert > C,$$

so the answer to Problem 67 is yes. This is Corollary 1.2 of the paper, the sign-valued case of its Theorem 1.1, which gives the same conclusion for every $f$ whose values have norm one in a real or complex Hilbert space. The paper is held as Tao 2016.

Argument. The proof has three steps. A Fourier-analytic reduction from the Polymath5 project replaces $f$ by a random completely multiplicative function $g$ with values in the unit circle whose discrepancy controls that of $f$. The author's logarithmically averaged form of the Elliott conjecture then forces such a $g$ to behave like a modulated Dirichlet character. An extension of a further Polymath5 argument shows that character-like functions still have unbounded discrepancy. The unit-norm hypothesis at every $n$ is necessary: a non-principal Dirichlet character of period $q$ has discrepancy at most $q$ but vanishes on the multiples of $q$.

Acceptance. Refereed: Discrete Analysis 2016, Paper No. 1, published 2016-02-28, following the preprint arXiv:1509.05363 of 2015-09-17 (six versions, the last of 2017-01-13). Reviewed: the site's curator, T. F. Bloom, records the question as true and proved by Tao in the problem's commentary (page last edited 2026-04-17, read 2026-10-07); the curator is independent of the author. The community database, which the claimant maintains, also lists the problem as proved; that listing is not independent review. The only formalization recorded is the statement in formal-conjectures, linked from the problem page; no kernel-checked proof is recorded, and this corpus has audited none.

Not covered. Erdős's further conjecture, stated in several of the problem's sources, that the maximum over $md\le x$ grows at least like $\log x$ is a separate question the paper leaves open. Its Example 1.4, the Borwein--Choi--Coons function, is a completely multiplicative $f:\mathbb{N}\to\{-1,+1\}$ whose discrepancy up to $N$ is comparable to $\log N$, so the conjectured order would be sharp, and its Example 1.5 is a unit-vector-valued $f$ whose discrepancy grows only like $\sqrt{\log N}$; the paper conjectures that $\sqrt{\log N}$ is best possible for Theorem 1.1, says it is unclear whether a $\pm1$ sequence can grow that slowly, and notes that its argument gives an explicit lower bound in principle, one likely far too weak to reach $\sqrt{\log N}$. The site records McNamara's lower bound of order $(\log\log x)^{1/484-o(1)}$ for the variant in which $m\le x$ and $d\le e^x$ (McNamara's 2021 UCLA dissertation, Chapter 4, entry [Mc21] on the problem page), and Erdős's question about multiplicative $f$. Neither bears on the standing of the question above.

Depends on. Nothing in this wiki; the result is the paper's own theorem.