Koukoulopoulos 2025 erdos s integer dilation approximation problem

../

theorem_1: Koukoulopoulos, Lamzouri and Lichtman's theorem that a discrete set of positive reals with positive upper logarithmic density has, for every epsilon > 0, distinct elements alpha, beta and a positive integer n with |n alpha - beta| < epsilon.

theorem_4_1: Koukoulopoulos, Lamzouri and Lichtman's weighted refinement of Behrend's theorem, bounding the sum of f(a)/a over a primitive set in a range [z/y, z] for a multiplicative f with 0 <= f <= tau_k.

---

Dimitris Koukoulopoulos, Youness Lamzouri, Jared Duker Lichtman, Erdős's integer dilation approximation problem and GCD graphs. arXiv:2502.09539v1 (13 February 2025); labels and pages on the result pages are those of this version. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2502.09539), every other right reserved.

The authors resolve Erdos's integer dilation approximation problem from 1948 in the case of his second hypothesis. Theorem 1 states that if A is a discrete subset of the positive reals with limsup_{x to infinity} (1/log x) sum over alpha in A cap [1,x] of 1/alpha > 0, then for every epsilon > 0 there exist distinct alpha, beta in A and a positive integer n with |n alpha - beta| < epsilon; a short remark upgrades this to infinitely many such pairs by removing found pairs and reapplying the theorem. The proof rests on the machinery of GCD graphs introduced by Koukoulopoulos and Maynard for the Duffin-Schaeffer conjecture, combined with a structure-versus-randomness dichotomy. The introduction places the result against Erdos's contrapositive formulation and the theory of primitive sets (Besicovitch's upper density 1/2 - epsilon examples, the Behrend and Erdos logarithmic-density-zero theorems, the Ahlswede-Khachatrian-Sarkozy refinement) and against Haight's 1988 work, which proved Theorem 1 in the special case where all ratios alpha/beta of distinct elements are irrational. A footnote records Erdos's 1997 prize offer for settling the problem. The paper does not treat Erdos's other condition, the divergence of sum_{alpha in A, alpha >= 2} 1/(alpha log alpha). For #143, Theorem 1 with epsilon = 1, read contrapositively, shows that a set A in (1,infinity) with |kx - y| >= 1 for all distinct x, y in A and integers k >= 1 (which, with k = 1, makes A 1-spaced and so discrete) has sum_{x in A, x < n} 1/x = o(log n); the paper does not treat the other assertion of #143, the convergence of sum_{x in A} 1/(x log x).

Source: https://arxiv.org/abs/2502.09539.

Bears on. #143

Results to transcribe.

• Theorem 1 (p. 2): if A is discrete in R_{>0} with limsup (1/log x) sum_{alpha in A cap [1,x]} 1/alpha > 0, then for every epsilon > 0 there are distinct alpha, beta in A and a positive integer n with |n alpha - beta| < epsilon; the Remark after it (p. 2) gives infinitely many such pairs, by reapplying the theorem to A minus the pairs already found.
• Theorem 4.1 (p. 22): for z >= y >= 2, a primitive set A of positive integers and a multiplicative f with 0 <= f <= tau_k, the sum of f(a)/a over a in A cap [z/y,z] is <<k (log y)/sqrt(L+1) exp(sum{p <= z} (f(p)-1)/p), where L = sum_{p <= y} f(p)/p; a refinement of Behrend's theorem used in the proof of Theorem 1.
• Method: GCD graphs, developed by Koukoulopoulos and Maynard for the Duffin-Schaeffer conjecture, in a structure-versus-randomness framework.

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