The board / additive combinatorics / #1191
Erdős Problem #1191
OPENdatabase updated 2026-04-04
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(A\subset\mathbb{N}\) be an infinite Sidon set. Is it true that \[\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^{1/2}=0?\] Here \(x\) runs over \(\mathbb{N}\) and the \(\liminf\) is taken in `ℝ≥0∞`, so that it is \(\infty\) rather than \(0\) when the sequence tends to infinity.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1191.Prize$1,000
Topicsadditive combinatorics, sidon sets
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1191
Attempts
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reduced both questions to exact monotone finite Golomb-ruler profiles, proved the quadratic profile through n=21, and constructed and exactly verified a critical-log-envelope ruler through n=680; no a
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