Walker 2022 integer sets large harmonic sum which

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Alexander Walker, Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions. arXiv:2203.06045 (2022).

The Erdos-Turan conjecture on arithmetic progressions asks whether every integer set with divergent harmonic sum contains arbitrarily long progressions; equivalently whether M_k, the supremum of harmonic sums over k-free sets, is finite for each k. The paper gives conditions under which Kempner sets K(S,b) - the nonnegative integers all of whose base-b digits lie in S - avoid k-term arithmetic progressions, and notes that the Baillie-Schmelzer algorithm evaluates their harmonic sums in polynomial time, which makes them well suited to large-scale search. Through such a search the author finds new lower bounds: the 4-free set K({0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47},55)+1 has harmonic sum 4.43975, improving the record for M_4 (already the simpler set K({0,1,2,4,5,7},11)+1 has harmonic sum 4.421746, beating the heuristic prediction of about 4.3 for the greedy set G_4), and an explicit Kempner set to base 77 is 10-free with harmonic sum 14.056, improving M_10 >= 13.5905 obtained from (G_7+3) union {1,2,3}. The paper thus supplies new lower bounds on these Erdos-Turan quantities relevant to erdosproblems.com/169, and reviews the general bounds M_k >= (1/2)k log 2 (Berlekamp) and M_k > (1-o(1))k log k (Gerver).

Source: https://arxiv.org/abs/2203.06045. The arXiv record (https://arxiv.org/abs/2203.06045, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #169

Results to transcribe.

• Lower bound for M_4: The 4-free Kempner set K({0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47},55)+1 has harmonic sum 4.43975, a new lower bound for M_4.
• Lower bound for M_10: An explicit Kempner set to base 77 is 10-free with harmonic sum 14.056, improving the previous bound M_10 >= 13.5905 from (G_7+3) union {1,2,3}.
• Simple 4-free example: K({0,1,2,4,5,7},11)+1 = {1,2,3,5,6,8,12,13,14,16,17,19,23,24,...} is 4-free with harmonic sum 4.421746, exceeding the heuristic estimate for the greedy 4-free set G_4.
• Kempner sets and progressions: Conditions are given under which a Kempner set K(S,b) avoids k-term arithmetic progressions, extending the connection developed in earlier work (Theorem 1.2: for b >= 3, if S, a proper subset of [0,b-1], is k-free mod b and contains 0, then K(S,b) is k-free); the Baillie-Schmelzer algorithm evaluates the harmonic sums of such sets in polynomial time.