Filaseta 2024 covering systems sum reciprocals moduli close

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theorem_1: Filaseta and Kalogirou's theorem that every finite covering system with distinct moduli, all exceeding 4, has reciprocal modulus sum at least 1 + exp(-3.363054 x 10^21), confirming the belief of Erdős and Selfridge.

theorem_2: Filaseta and Kalogirou's theorem that a finite distinct covering system whose 3-smooth moduli leave uncovered a density Delta in (0, 1/12) has reciprocal modulus sum at least 1 + exp(-(5.846 x 10^20 - 1.242 x 10^19 log Delta)/Delta), whatever its minimum modulus.

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Michael Filaseta, Alexandros Kalogirou, Covering systems with the sum of the reciprocals of the moduli close to 1. arXiv preprint (2024). arXiv:2407.15280.

The copy read for this card is arXiv:2407.15280v1 (30 pages, dated 23 July 2024), the only version on the arXiv record. The paper answers a problem of Davenport, reported by Erdos in 1952, by proving the belief Erdos and Selfridge stated in 1973. Theorem 1 (PDF p. 2): every finite distinct covering system whose minimum modulus exceeds 4 has reciprocal modulus sum at least 1+exp(-3.36305410^21). The authors note that what matters is the density Delta of integers left uncovered by the congruences with 3-smooth moduli: a minimum modulus above 4 forces Delta > 1/12, the proof of Theorem 1 gives the same bound whenever Delta >= 1/12, and Theorem 2 (PDF p. 3) gives, for a finite distinct covering system with Delta in (0, 1/12), the lower bound 1+exp(-(5.84610^20-1.24210^19log(Delta))/Delta). The proofs use the distortion method of Balister, Bollobas, Morris, Sahasrabudhe and Tiba together with ideas of E. Lewis. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2407.15280), every other right reserved.

Source: https://arxiv.org/abs/2407.15280.

Results.

• Theorem 1 (p. 2): minimum modulus above 4 forces reciprocal sum at least 1+exp(-3.363054*10^21); the page also records the remark on Delta >= 1/12 (p. 3).
• Theorem 2 (p. 3): the Delta-dependent bound for Delta in (0, 1/12).

Read status. Claims checked for Theorems 1 and 2, the definition of Delta and the remarks on pp. 2-3; the reduction in Section 3 (pp. 6-12) was read for its structure, and Lemmas 1 to 4 and their proofs (pp. 6-26) were not checked.

Bears on.

• Problem 273: a necessary condition, not an answer. A distinct covering system with all moduli of the form p-1, p >= 5, that avoids the modulus 4 has reciprocal sum at least 1+exp(-3.363054*10^21) by Theorem 1; one that uses the modulus 4 gets the same bound when Delta >= 1/12 by the remark on p. 3 and the bound of Theorem 2 when 0 < Delta < 1/12, and nothing when Delta = 0. A July 2026 research note recorded as claimed on its claim page derives a bound of this size for all such covering systems from the paper's framework; that derivation is the note's own.

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