\(B_h[g]\) modular sets from \(B_h\) modular sets. (English) Zbl 1499.11089
Summary: A set of positive integers \(A\) is called a \(B_h[g]\) set if there are at most \(g\) different sums of \(h\) elements from \(A\) with the same result. This definition has a generalization to abelian groups and the main problem related to this kind of sets is to find \(B_h[g]\) maximal sets, i.e., those with larger cardinality. We construct \(B_h[g]\) modular sets from \(B_h\) modular sets using homomorphisms and analyze the constructions of \(B_h\) sets by Bose and Chowla, Ruzsa, and Gómez and Trujillo look at for the suitable homomorphism that allows us to preserve the cardinal of these types of sets.
MSC:
11B50 | Sequences (mod \(m\)) |
12E20 | Finite fields (field-theoretic aspects) |
20K01 | Finite abelian groups |
20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
References:
[1] | R. C. Bose, An affine analogue of Singer’s theorem, J. Indian Math. Soc. (N.S.) 6 (1942), 1-15. · Zbl 0063.00542 |
[2] | R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Comment. Math. Helv. 37 (1962/1963), 141-147. · Zbl 0109.03301 |
[3] | I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith. 65 (1993), 259-282. · Zbl 1042.11525 |
[4] | J. Singer, A theorem infinite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377-385. · JFM 64.0972.04 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.