Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
27 January 2020
In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.
27 January 2020
In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.
27 January 2020
In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.
27 January 2020
In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.
References
S. Baier, A note on P-sets, Integers, 4 (2004), A13, 6 pp.
Baier S.: On the large sieve with sparse sets of moduli. J. Ramanujan Math. Soc., 21, 279–295 (2006)
S. Baier and R. K. Singh, Large sieve inequality with power moduli for function fields, ar**v:1802.03131 [math.NT] (2018).
Baier S., ZhaoL L.: Large sieve inequality with characters for powerful moduli. Int. J. Number Theory, 1, 265–279 (2005)
E. Bombieri, Le grand crible dans la thorie analytique des nombres, 2nd ed., Astérisque, 18, Société Mathématique de France (Paris, 1987).
Elsholtz C., Planitzer S.: On Erdős and Sárközy’s sequences with Property P. Monatsh. Math., 182, 565–575 (2017)
Erdős P., Sárközy A.: On divisibility properties of integers of the form a + a. Acta Math. Hungar., 50, 117–122 (1987)
Hsu C.-N.: A large sieve inequality for rational function fields. J. Number Theory, 58, 267–287 (1996)
H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag (Berlin–New York, 1971).
J. Rivat and A. Sárközy, On arithmetic properties of products and shifted products, in: Analytic Number Theory, Springer (Cham, 2015), pp. 345–355.
Schoen T.: On a problem of Erdős and Sárközy. J. Comb. Theory Ser. A, 94, 191–195 (2001)
Zhao L.: Large sieve inequality with characters to square moduli. Acta Arith., 112, 297–308 (2004)
Acknowledgement
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baier, S., Bansal, A. & Singh, R.K. Divisibility problems for function fields. Acta Math. Hungar. 156, 435–448 (2018). https://doi.org/10.1007/s10474-018-0853-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0853-4