Dombi, Gergely Additive properties of certain sets. (English) Zbl 1014.11009 Acta Arith. 103, No. 2, 137-146 (2002). 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. Reviewer: Jean-Paul Allouche (Orsay) Cited in 10 ReviewsCited in 37 Documents MSC: 11B34 Representation functions 11B83 Special sequences and polynomials 11B85 Automata sequences Keywords:number of representations; Rudin-Shapiro sequence; additive properties of sequences of integers; Thue-Morse sequence Citations:Zbl 0588.10056 PDFBibTeX XMLCite \textit{G. Dombi}, Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009) Full Text: DOI Online Encyclopedia of Integer Sequences: Number of compositions of n into three parts, using only natural numbers not in A007283. Lexicographically least counterexample to Dombi’s conjecture.