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Hough 2019 covering systems restricted divisibility
Hough, Robert D. and Nielsen, Pace P., Covering systems with restricted divisibility. Duke Math. J. 168 (2019), no. 17, 3261-3295. DOI: 10.1215/00127094-2019-0058. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1703.02133), every other right reserved. The copy read for this card is the arXiv v2 PDF (8 August 2018).
Theorem 1 states that every distinct covering system of congruences (distinct moduli, all greater than 1) has some modulus divisible by 2 or by 3. This is negative progress on the Erdős-Selfridge problem of whether a distinct covering system with all moduli odd exists, and answers a question raised in the literature; earlier work had only shown that an odd squarefree distinct covering system must involve at least 18 primes (Simpson-Zeilberger) or 22 primes (Guo-Sun). The proof works probabilistically in Z/QZ with Q the lcm of the moduli: it gives a positive lower bound for the density of the uncovered set R = Z minus the union of the residue classes, although quantitatively it estimates related quantities. It sieves in stages, grouping moduli by prime factors in successive ranges: the first stage uses a Shearer-type theorem (Theorem 2), later stages a Lovász-type theorem (Theorem 4) inside 'good' fibres modulo the partial lcm Q_i, each stage weighted by a probability measure on Z/Q_iZ chosen so that most fibres are good. The paper also recalls the first author's result that the least modulus of a distinct covering system is at most 10^16. For problem 7 (odd covering systems) it rules out every distinct covering system (distinct moduli, all greater than 1) whose moduli avoid both 2 and 3.
Source: https://arxiv.org/abs/1703.02133.
Bears on. #7
Results to transcribe.
- Theorem 1 (p. 1 of arXiv v2): "Every distinct covering system of congruences has a modulus divisible by either 2 or 3."
- Density set-up (Section 2): Writes P(R) for the density of the uncovered set in Z/QZ and reduces Theorem 1 to a positive lower bound for that probability.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.