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Additive properties of certain sets. (English) Zbl 1014.11009

The author addresses interesting questions on additive properties of sequences of integers. If \(A= \{a_1< a_2<\dots\}\) is an infinite sequence of integers, define \(r(k,A,n)\) to be the number of solutions of \(x_1+\cdots+ x_k= n\), with all \(x_i\)’s in \(A\). It was proved by P. Erdős, A. Sárközy and V. Sós [Lect Notes Math. 1122, 85-104 (1985; Zbl 0588.10056)] that, if \(r(2,A,n)\) is increasing for \(n\) large enough, then \(A\) must contain all integers from some point on. Here the author constructs a sequence \(A\) such that \(r(k,A,n)\) is increasing for every \(k> 4\), and for all \(n> n_0(k)\), and the natural density of the sequence \(A\) is \(\frac 12\). Another additive problem is addressed, using the famous Thue-Morse sequence. The paper ends with five open problems.

MSC:

11B34 Representation functions
11B83 Special sequences and polynomials
11B85 Automata sequences

Citations:

Zbl 0588.10056
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