Addendum to “Structure Theorem for Multiple Addition”

✍ Scribed by Vsevolod F. Lev

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 283 KB
• Volume: 65
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2128

✦ Synopsis

In our earlier paper we gave a sharp lower-bound estimate for the cardinality of hA=A+ } } } +A (A being a finite set of integers). It appears that certain applications require this estimate to be generalized for the case of distinct summands.

Below we obtain such a generalization by estimating the cardinality of A 1 + } } } A h (A i being distinct finite sets of integers).

1997 Academic Press

In the present paper, we prove the following theorem, generalizing both [4, Theorem 1] and [5, Theorem 2].

Theorem. Let A 1 , ..., A h [0; l] be nonempty sets of integers, and assume that 0, l # A h and gcd(A h )=1. For i=1, ..., h define

Let A [0; l ] be a set of integers of the cardinality n 2 and satisfying 0, l # A, gcd(A)=1. For h 1 , h 2 1 define h=h 1 +h 2 and write

Then evidently |h 1 A&h 2 A| =|A 1 + } } } +A h |, and using induction exactly in the same way as in , one can derive the following corollary.