Erdős Problem #995
OPENdatabase updated 2025-09-07
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The partial sum \(S_N(\alpha) = \sum_{k < N} f(\{\alpha n_k\})\).
Erdős Problem 995: For every lacunary sequence \((n_k)\) of integers and every \(f \in L^2([0,1])\) with \(\int_0^1 f = 0\), is it true that for almost all \(\alpha\), \[\sum_{k < N} f(\{\alpha n_k\}) = o\!\left(N \sqrt{\log\log N}\right)?\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/995.Prizenone
Topicsanalysis, discrepancy
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/995
Attempts
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LIVE PAGE STATUS OPEN (0 formal claimed proofs; no current-worker or interested markers), but its 21–22 April 2026 discussion contains a claimed solution—arXiv:2604.18535 is said to give negative answ
SKIPPED