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Let be a real number, not an integer. The density of integers for which is .
Source: erdosproblems.com/1149
An accepted solution exists. The statement is true.
Proved. Theorem 1 of Bergelson and Richter ([BeRi17], a chapter of a Springer Festschrift volume, arXiv:1611.08044) gives density for the integers coprime to for every Hardy-field meeting its growth conditions, with a non-integer among them; the site's curator, Thomas Bloom, records this as the proof. The claim page Bergelson and Richter 2017 records the acceptance. Earlier, Delmer and Deshouillers (2002) proved the statement for every non-integer , Delmer and Deshouillers 2002, and Lambek and Moser (1955) proved the exponents for integers , Lambek and Moser 1955. Bergelson and Richter's introduction credits Lambek and Moser with all , which is more than their paper states.