Problem 995

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claims/: The 1 claim page of Problem 995, one per claimant's result; the problem's standing derives from them.

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Statement. Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$,

$$\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).$$

For example, is it true that, for almost all $\alpha$,

$$\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?$$

Status. OPEN on erdosproblems.com; the site's commentary records Erdős's bounds and does not name a later result. One pending partial claim answers the example question in the negative: Boon Suan Ho's preprint of 20 April 2026, announced in the problem's thread on 21 April 2026, Ho 2026. No claim had been filed on the site's proof-claims tab.

Source. erdosproblems.com/995, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #995, https://www.erdosproblems.com/995.

References.

• [Er49d] Erdős, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51-56.
• [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation asks two things: to estimate the almost-everywhere growth of $\sum_{k\le N}f(\{\alpha n_k\})$ for a lacunary sequence and $f\in L^2$, and, as an example, whether the sums are $o(N\sqrt{\log\log N})$. Theorem 1.6 of Ho's v2 (Theorem 1.5 of v1) at $p=2$ answers the example question no, with a mean-zero $f\in L^2$ and a lacunary sequence whose sums exceed $N(\log N)^{1/2-\varepsilon}$ infinitely often almost everywhere for every $\varepsilon>0$; the claim is pending on its claim page; the site labels the problem OPEN and records no proof claim. Together with Erdős's upper bound $o(N(\log N)^{1/2+\varepsilon})$, the worst case over lacunary sequences and square-integrable $f$ grows like $N(\log N)^{1/2+o(1)}$, the reading on which the author suggested the growth question is answered, while noting that this depends on interpretation, and a thread participant said it may qualify as a full solution pending confirmation; the problem names no target for "estimate the growth", so this corpus records that reading here and claims only the example question on the claim page. No independent review of the proofs is recorded.

Known Results

Erdős [Er49d] constructed a lacunary sequence and $f\in L^2([0,1])$ such that, for every $\epsilon>0$ and almost all $\alpha$, the sums $\sum_{k\le N}f(\{\alpha n_k\})$ exceed $N(\log\log N)^{1/2-\epsilon}$ infinitely often, and proved that for every lacunary sequence and every $f\in L^2$ the sums are $o(N(\log N)^{1/2+\epsilon})$ for almost all $\alpha$; in [Er64b] he thought the lower bound closer to the truth.

Ho (arXiv:2604.18535, Theorem 1.6 and Remark 7.1 of v2) gives, for each $2\le p<\infty$, a mean-zero $f\in L^p$ and a lacunary sequence with $n_{j+1}/n_j\ge2$ whose partial sums exceed $N(\log N)^{1/p-\varepsilon}$ infinitely often almost everywhere, against the elementary upper bound $O(N(\log N)^{1/p+\varepsilon})$ valid for every increasing sequence; at $p=2$ this refutes the $o(N\sqrt{\log\log N})$ example and shows Erdős's upper bound sharp up to the $\varepsilon$ gap. See the claim page and the library card.

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_1949_strong_law_large_numbers
• erdos_1949_strong_law_large_numbers / remark_p52
• erdos_1949_strong_law_large_numbers / theorem_1
• ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike
• erdos_1964_problems_results_diophantine_approximations