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Claim. Let $f$ belong to a Hardy field and satisfy the paper's two growth conditions: (A) $\log(t)\log_4(t)\prec f(t)$, and (B) $t^{j-1}\prec f(t)\prec t^j$ for some $j\in\mathbb{N}$, where $\log_4$ is the fourth iterated logarithm and $g\prec h$ means $h(t)/g(t)\to\infty$. Then the set of $n\ge1$ with $\gcd(n,\lfloor f(n)\rfloor)=1$ has natural density $6/\pi^2$. This is Theorem 1 of V. Bergelson and F. K. Richter, On the density of coprime tuples of the form $(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)$, where $f_1,\ldots,f_k$ are functions from a Hardy field, arXiv:1611.08044 (v1, 24 November 2016, the page name's date; v2, 20 May 2017), published in Number Theory -- Diophantine Problems, Uniform Distribution and Applications (Festschrift for Robert F. Tichy), Springer, 2017, pp. 109--135, DOI 10.1007/978-3-319-55357-3_5. The paper lists $f(t)=t^c$ for non-integer $c>0$ among the functions meeting (A) and (B) ($t^c$ with $j-1<c<j$ satisfies both), which is the statement of Problem 1149, so the problem's assertion is proved. The paper remarks that (A) is sharp, citing Erdős and Lorentz for the failure of the theorem at $f(t)=\log(t)\log_4(t)$, and proposes a conjectural replacement (B') for (B), that $f(t)\prec t^j$ for some $j$ and $|f(t)-p(t)|\succ\log(t)$ for every rational polynomial $p$; (B') is not a hypothesis of the theorem. Theorem 2 is the $k$-tuple version with density $1/\zeta(k+1)$ under a separation condition (C), $f_{i+1}/f_i\succ\log_2^4(t)$. The proof runs through differential inequalities for Hardy-field functions, van der Corput's estimates for the resulting exponential sums, discrepancy bounds, and an inclusion-exclusion over divisors. The source card digests the paper.

Acceptance. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED, last edited 23 January 2026, and the commentary states that the assertion is true and attributes the proof to Bergelson and Richter. Refereed: the publication is a chapter of an edited Springer volume (its Crossref record types it a book chapter), and its acknowledgements thank the anonymous referees and the editor who handled the submission. This page rests on no review of its own.

Formalization. None: the site records no formalized statement, and formal-conjectures had no 1149.lean on 2026-09-05.

Scope. Full. The theorem covers every Hardy-field $f$ meeting (A) and (B), of which the problem's $n^\alpha$ is one instance. The earlier results that the paper's introduction records are not part of this claim: Watson (Canadian J. Math. 5, 1953) proved the density $6/\pi^2$ for the linear function $f(n)=\alpha n$ with $\alpha$ irrational, Lambek and Moser (Canad. J. Math. 7, 1955) for $n^c$ with $0<c<1$ in Bergelson and Richter's attribution, though that paper states only $c=1/k$ for integers $k\ge2$, and Delmer and Deshouillers (Period. Math. Hungar. 45, 2002) for every non-integer $c>0$. Those two results have their own claim pages, Lambek and Moser 1955 and Delmer and Deshouillers 2002.