Koukoulopoulos 2020 duffin schaeffer conjecture

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corollary_3: For psi from the positive integers to [0,1/2], the set of alpha in [0,1] with infinitely many coprime solutions of |alpha - a/q| <= psi(q)/q has Hausdorff dimension min(s,1), where s is the infimum of the beta >= 0 for which the sum of phi(q)(psi(q)/q)^beta converges.

theorem_1: For every function psi from the positive integers to the nonnegative reals with the sum of psi(q) phi(q)/q divergent, almost every alpha in [0,1] lies within psi(q)/q of infinitely many fractions a/q with a and q coprime.

theorem_2: For every function psi from the positive integers to the nonnegative reals, the set of alpha in [0,1] lying within psi(q)/q of infinitely many fractions a/q with 0 <= a <= q, not necessarily reduced, has measure 0 or 1 according as the sum of psi*(q) = phi(q) sup{psi(n)/n : q | n} converges or diverges.

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Koukoulopoulos, Dimitris and Maynard, James, On the Duffin-Schaeffer conjecture. Ann. of Math. (2) 192 (2020), no. 1, 251--307. DOI 10.4007/annals.2020.192.1.5.

For ψ : N → R_{≥0}, let A be the set of α ∈ [0,1] for which |α - a/q| ≤ ψ(q)/q has infinitely many solutions in coprime integers a and q. Theorem 1 proves the Duffin-Schaeffer conjecture: if sum_q ψ(q)φ(q)/q = ∞ then A has Lebesgue measure 1. Duffin and Schaeffer posed this in 1941 (it is Problem 46 in Montgomery's lectures), and no monotonicity of ψ is assumed; the converse implication, measure 0 when the series converges, is the easy direction of the Borel-Cantelli lemma, recorded on p. 2. Theorem 2 deduces Catlin's conjecture for not-necessarily-reduced approximations: with ψ*(q) = φ(q) sup{ψ(n)/n : q | n}, the set K of α ∈ [0,1] with infinitely many solutions (a, q) with 0 ≤ a ≤ q has measure 0 or 1 according as sum ψ*(q) converges or diverges, which the paper calls an extension of Khinchin's theorem. Corollary 3, Theorem 1 combined with a result of Beresnevich and Velani, gives the Hausdorff dimension of A as min(s, 1) when ψ takes values in [0, 1/2], s the infimum of the β ≥ 0 with sum φ(q)(ψ(q)/q)^β < ∞. The method is combinatorial and graph-theoretic: a second-moment argument with Gallagher's zero-one law reduces Theorem 1 to bounding the overlaps of the sets A_q, and the authors bound them by studying bipartite 'GCD graphs' with a density-increment and compression argument.

Source: https://arxiv.org/abs/1907.04593. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1907.04593), every other right reserved.

The copy read for this card is the arXiv preprint, arXiv:1907.04593v3.

Results. Labels and pages are those of arXiv:1907.04593v3.

• Theorem 1 (p. 2): the Duffin-Schaeffer conjecture; if sum ψ(q)φ(q)/q diverges, almost every α ∈ [0,1] has infinitely many coprime solutions of |α - a/q| ≤ ψ(q)/q.
• Theorem 2 (p. 3): Catlin's conjecture; for approximations that need not be reduced, the measure is 0 or 1 according as sum ψ*(q) converges or diverges.
• Corollary 3 (p. 4): for ψ with values in [0, 1/2], the set A has Hausdorff dimension min(s, 1).

Read status. Claims checked: the statements of Theorems 1 and 2 and Corollary 3 were read clause by clause; the deduction of Theorem 2 in Section 2 was followed, and the proof of Theorem 1 was read for structure only.

Bears on. #999 (Theorem 1 is the problem's divergence half and the paper's display (1.5) its convergence half, both for real-valued ψ ≥ 0, α ∈ [0,1] and the non-strict inequality ≤; the problem states the equivalence for f : N → N with the strict inequality <, which the paper does not address)

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.