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Updated Oct 8

Problem 1147

claims/: The 1 claim page of Problem 1147, one per claimant's result; the problem's standing derives from them.

Statement. Let $\alpha>0$ be an irrational number. Is the set

A=\left\{ n\geq 1: \| \alpha n^2\| < \frac{1}{\log n}\right\},

where $\|\cdot\|$ denotes the distance to the nearest integer, an additive basis of order $2$ ?

Status. Disproved. Konieczny [Ko16b] proves that the set is not an additive basis of order two for almost every $\alpha>0$ , and explicitly for $\alpha=\sqrt2$ , with any $\epsilon(n)\to0$ in place of $1/\log n$ ; for thresholds decaying slowly enough (an $\alpha$ -dependent rate that the paper does not relate to $1/\log n$ ) the set is a basis of order two for uncountably many exceptional $\alpha$ and of order three for every irrational $\alpha$ . The accepted claim is Konieczny.

Source. erdosproblems.com/1147, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1147, https://www.erdosproblems.com/1147.

References.

[Ko16b] Konieczny, Jakub, Sets of recurrence as bases for the positive integers. Acta Arith. (2016), 309-338.

Formalization. Statement in formal-conjectures. The community database at teorth/erdosproblems records the problem's formal status as Lean; the two third-party Lean formalizations of Konieczny's $\sqrt2$ counterexample, neither built by this corpus, are linked from the claim page.

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Linked library material

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Linked from (10)

Konieczny's quadratic recurrence sets that are not bases of order two problems/additive_bases/E1147/claims Additive Bases and Sidon Sets additive_bases/konieczny_2016_sets_recurrence_as_bases_positive_integers Lemma 1.2 (p. 5): odd N_i with N_i alpha = m_i/k + gamma_i/(k N_i) and an accumulation point gamma, |gamma| < 1, lie outside 2A for infinitely many i when eps(n) -> 0, unless gamma + k n^2 alpha is an integer for some n Proposition 1.3 (p. 6): {n : ||sqrt2 n^2|| < eps(n)} is not a basis of order 2 if eps(n) -> 0 or eps(n) <= eps_0 < (1 - 1/(4 sqrt2))/4 Proposition 2.9 (p. 18) and Theorem (A2 reiterated) (p. 12): for uncountably many alpha, {n : ||alpha n^2|| < eps(n)} is a basis of order 2 for slowly decaying eps Question 1 (p. 2): is the set of n with ||sqrt2 n^2|| <= 1/log n a basis of order 2? Answered no Theorem A (p. 2): for irrational alpha, {n : ||alpha n^2|| <= eps(n)} is an almost basis of order 2 and a basis of order 3 for slowly decaying eps, and for almost all alpha not a basis of order 2 when eps(n) -> 0 Theorem (A4 reiterated) (p. 4) with Propositions 1.4 to 1.6: for almost all alpha and all quadratic irrationals, {n : ||alpha n^2|| < eps(n)} is not a basis of order 2 when eps(n) -> 0

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