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Erdős covering systems


A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most 1016.

The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough’s method, which was recently used to answer several other questions about covering systems. We hope that this technique, which we call thedistortion method, will have many further applications in other combinatorial settings.

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  1. 1.

    P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, On the Erdős covering problem: the density of the uncovered set (submitted), arXiv:1811.03547

  2. 2.

    P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The Erdős–Selfridge problem with square-free moduli (submitted), arXiv:1901.11465

  3. 3.

    Erdős, P.: On integers of the form \(2^k + p\) and some related problems. Summa Brasil. Math. 2, 113–123 (1950)

    MathSciNet  Google Scholar 

  4. 4.

    Filaseta, M., Ford, K., Konyagin, S., Pomerance, C., Yu, G.: Sieving by large integers and covering systems of congruences. J. Amer. Math. Soc. 20, 495–517 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Hough, R.: Solution of the minimum modulus problem for covering systems. Ann. Math. 181, 361–382 (2015)

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to R. Morris.

Additional information

To Endre Szemerédi, a truly extraordinary mathematician, a wonderful friend, and a great inspiration, on his 80th birthday

The first two authors were partially supported by NSF grant DMS 11855745.

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Balister, P., Bollobás, B., Morris, R. et al. Erdős covering systems. Acta Math. Hungar. 161, 540–549 (2020).

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Key words and phrases

  • covering system
  • distortion method
  • hyperplane

Mathematics Subject Classification

  • 05B40
  • 05C70
  • 11A07
  • 11B25