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Claim. Let $s_1<s_2<\cdots$ be the squarefree numbers. For every real $0\le\alpha<11/3$ there is a constant $B(\alpha)>0$ with $\sum_{s_{n+1}\le x}(s_{n+1}-s_n)^\alpha\sim B(\alpha)\,x$, so the limit that Problem 145 asks about exists for every $\alpha$ in that range. The source is M. N. Huxley, Moments of differences between square-free numbers, Chapter 11 of G. R. H. Greaves, G. Harman and M. N. Huxley (eds.), Sieve Methods, Exponential Sums, and their Applications in Number Theory (Cardiff, 1995), London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press (1997), pp. 187--204 (the publisher's record; Chan's reference list prints the pages as 235--253). The site's commentary credits the result to the volume's three editors under its key [GHH97]; the chapter is Huxley's alone, and Chan's 2023 paper, which refines its method, cites it as Huxley's and states its range as $0\le\gamma<11/3$. The site prints the range as $\alpha\le11/3$; this page follows Chan's statement, which the formal-conjectures variant erdos_145.variants.lt_eleven_thirds also uses. The argument classifies the gaps $s_{n+1}-s_n\ge H+1$ by whether the run of non-squarefree integers contains a multiple of $p^2$ for a large prime $p$ or many multiples of squares of mid-sized primes, and counts the latter through lattice points close to a curve.

Covers. Every exponent $0\le\alpha<11/3$; the exponents up to $3$ are already covered by Hooley 1973, and the exponents $11/3\le\alpha<3.75$ by Chan 2023. Huxley's 2000 paper The rational points close to a curve II (Acta Arith. 93 (2000), 201--219) widened the range to $0\le\alpha<59/16$ before Chan's result, as Chan's introduction records; the site does not credit it and it is subsumed, so it has no page.

Standing. Claimed. The volume is a conference proceedings, and no record names its refereeing, so refereed is not listed; the site labels the problem OPEN (page last edited 19 October 2025), so its curator's remark crediting the range is commentary on an open problem and not reviewed evidence. The corpus holds no card for the chapter, has not reproved the theorem and awards no tier of its own; Chan's refereed paper covers a wider range, so the problem's standing does not rest on this page.

Depends on. Nothing in this wiki; the claim rests on the cited chapter.