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On the problems associated with sum of dilates. (English) Zbl 07820494

Let \(A\) be a subset of integers and \(r,s\) be positive integers. In this paper under review, the authors study the size of the set \(r\cdot A+ s \cdot A=\{ra+sb: a,b \in A\}\).
G. A. Freiman et al. [Eur. J. Comb. 40, 42–54 (2014; Zbl 1315.11089)] proved that \(|A+ k \cdot A|\geq 4|A|-4\) whenever \(k\geq 3\). In Section 2, the authors prove that if \(p\) is a prime and \(k>p^2\), then \(|p \cdot A+ k \cdot A|\geq 4|A|-4\) under the additional assumption that the gaps between consecutive elements in \(A\) lie in the interval \([2p+1,k]\). The proof is elementary.
Inverse problems related to sums of dilates are also studied, motivated by the celebrated Freiman’s \(3k-4\) theorem, which states that if \(|A+A|\leq 3k-4\) with \(|A|=k\), then \(A\) is contained in an arithmetic progression of length at most \(2k-3\). In Section 3, the authors prove that if \(|A+3 \cdot A|<5k-6\) with \(|A|=k\), then \(A\) is contained in an arithmetic progression of length at most \(2k-3\) under the additional assumption that \(A \pmod 3=\{0,1,2\}\). The proof is elementary and relies on well-known results by V. F. Lev and P. Y. Smeliansky [Acta Arith. 70, No. 1, 85–91 (1995; Zbl 0817.11005)] and Y. V. Stanchescu [Acta Arith. 75, No. 2, 191–194 (1996; Zbl 0841.11008)].

MSC:

11B13 Additive bases, including sumsets
11P70 Inverse problems of additive number theory, including sumsets
11B30 Arithmetic combinatorics; higher degree uniformity
Full Text: DOI

References:

[1] Cilleruelo, J.; Silva, M.; Vinuesa, C., A sumset problem, J. Comb. Number Theory, 2, 79-89, 2010 · Zbl 1245.11029
[2] Freiman, G. A.; Herzog, M.; Longobardi, P.; Maj, M.; Stanchescu, Y. V., Direct and inverse problems in additive number theory and in non-abelian group theory, Eur. J. Combin., 40, 42-54, 2014 · Zbl 1315.11089 · doi:10.1016/j.ejc.2014.02.001
[3] Nathanson, M. B., Additive Number Theory: Inverse Problems and the Geometry of Sumsets, 1996, New York, NY: Springer, New York, NY · Zbl 0859.11003 · doi:10.1007/978-1-4757-3845-2
[4] Cilleruelo, J.; Hamidoune, Y. O.; Serra, O., On sums of dilates, Combin. Probab. Comput., 18, 871-880, 2009 · Zbl 1200.11007 · doi:10.1017/S0963548309990307
[5] Du, S. S.; Cao, H. Q.; Sun, Z. W., On a sumset problem for integers, Electron. J. Combin., 21, 0, 2014 · Zbl 1308.11010 · doi:10.37236/2801
[6] Freiman, G. A., Foundations of a Structural Theory of Set Addition, 1973, Providence, RI: Am. Math. Soc., Providence, RI · Zbl 0271.10044
[7] Lev, V. F.; Smeliansky, P. Y., On addition of two distinct sets of integers, Acta Arith., 70, 1, 85-91, 1995 · Zbl 0817.11005 · doi:10.4064/aa-70-1-85-91
[8] Stanchescu, Y. V., On addition of two distinct sets of integers, Acta Arith., 75, 2, 191-194, 1996 · Zbl 0841.11008 · doi:10.4064/aa-75-2-191-194
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