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Let A be a set of non-negative integers. In our previous paper for h # Z, h 2 we estimated from below the cardinality of hA and this allowed us to obtain sharp estimates for such G, that every integer g G may be represented by a sum of elements of A (the linear diophantine problem of Frobenius). However, ``not only the number of elements of hA is important, but also the way they are situated'' (G. Freiman). In this paper we show that hA always contains long continuous chains of integers and derive an estimation for the number of summands required for representation of g of the above mentioned type. 1997 Academic Press 1. NOTATION If the opposite is not stated explicitly, we will use the following notations. A will denote a set of n= |A| 3 integers, contained in the segment [0; l] and satisfying 0, l # A, gcd(A)=1. For an integer h 1, by hA we denote the h-fold sumset A+ } } } +A (h times), that is the set of all the integers representable by a sum of h elements of A. By h } A we denote the set of all those integers, representable by a sum of exactly h distinct elements of A: We write [u; v] hA meaning that hA contains all the integers from the segment [u; v]. If there exists such a

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Author

Vsevolod F Lev

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-314X)

Published

1997

Language

EN

Field

Mathematics (Physical Sciences)