Biró, András Divisibility of integer polynomials and tilings of the integers. (English) Zbl 1088.11016 Acta Arith. 118, No. 2, 117-127 (2005). If \(A\) and \(B\) are sets of integers and every integer can be written as \(a+b\), \(a\in A\), \(b \in B\), in a unique way, we say that \(A\) tiles the integers when translated by \(B\). If this is the case and we assume that \(A\) is finite then it is well known that \(B\) is a periodic set and D. J. Newman [“Tesselations of the integers”, J. Number Theory 9, 107–111 (1977; Zbl 0348.10038)] first proved the upper bound \(2^D\) for the least period of \(B\), where \(D\) is the diameter of \(A\).The reviewer [Electron. J. Comb. 10, No. 1, R22 (2003; Zbl 1107.11016)] and I. Z. Ruzsa [Appendix to R. Tijdeman, Bolyai Soc. Math. Stud. 15, 381–405 (2006; Zbl 1103.68103)] have improved this to roughly \(2^{\sqrt{D \log D}}\) and in this paper the bound is further improved to \(2^{D^{1/3+\varepsilon}}\), for any positive \(\varepsilon\). Reviewer: Mihail Kolountzakis (Iraklio) Cited in 1 ReviewCited in 9 Documents MSC: 11C08 Polynomials in number theory 11B75 Other combinatorial number theory Citations:Zbl 0348.10038; Zbl 1107.11016; Zbl 1103.68103 × Cite Format Result Cite Review PDF Full Text: DOI