Divisibility problems for function fields
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We investigate three combinatorial problems considered by Erdős, Rivat, Sárközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.
Key words and phrasesdivisibility problem function field large sieve
Mathematics Subject Classification11B75 11N36 11L40 11P99
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We thank Igor Shparlinski for making us aware of Theorems 1.2 and 1.3 and suggesting to consider them in the function field setting.
- 1.S. Baier, A note on P-sets, Integers, 4 (2004), A13, 6 pp.Google Scholar
- 3.S. Baier and R. K. Singh, Large sieve inequality with power moduli for function fields, arXiv:1802.03131 [math.NT] (2018).
- 5.E. Bombieri, Le grand crible dans la thorie analytique des nombres, 2nd ed., Astérisque, 18, Société Mathématique de France (Paris, 1987).Google Scholar
- 9.H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag (Berlin–New York, 1971).Google Scholar
- 10.J. Rivat and A. Sárközy, On arithmetic properties of products and shifted products, in: Analytic Number Theory, Springer (Cham, 2015), pp. 345–355.Google Scholar