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Given let be the set of multiples of . Find a necessary and sufficient condition on for to have density .
Source: erdosproblems.com/691
No claim settles this problem.
Open, the site's label. The site credits Tenenbaum's 1996
theorem. For block sequences whose consecutive ratios lie between two
constants above and whose blocks have relative length , it
proves Erdős's threshold conjecture with critical exponent
(claim page (Tenenbaum, 1996)),
an accepted partial claim with refereed evidence. It does not answer the
general question, so the derived standing is open with claim none. A
thread note of 17 April 2026 with a Lean formalization, both produced with
GPT-5.4 Pro, proves the classical Davenport--Erdős criterion: has
density exactly when the densities of the multiples of tend
to . On 18 April 2026 its author recast it as an exposition of that known
fact (equation (1.3) of Hall and Tenenbaum), so it has no claim page.