Valkó, Benedek On irregularities of sums of integers. (English) Zbl 0946.11006 Acta Arith. 92, No. 4, 367-381 (2000). Let \(A\subseteq\{1,\dots,N\}\) and \(\eta=|A|/N\). K. F. Roth [Acta Arith. 9, 257-260 (1964; Zbl 0125.29601)] proved that unless \(A\) is empty or equal to \(\{1,\dots,N\}\) it cannot be well-distributed simultaneously among and within all congruence classes. In the first part of the present paper the author proves a similar statement about the distribution of sums \(a_1+a_2\) where \(a_1,a_2 \in A\).In the second part of the paper the corresponding problem in residue classes is studied. Consider a subset \(A\subset\{0, \dots, N-1\}\) of the residues modulo \(N\). The sums \(a_1+a_2\) with \(a_1,a_2 \in A\) are well-distributed if the numbers of sums in each residue class are about the same. However, these sums cannot be too well-distributed this way. Reviewer: R.F.Tichy (Graz) MSC: 11B25 Arithmetic progressions 11K38 Irregularities of distribution, discrepancy Keywords:irregularities of sums of integers; congruence classes; residue classes Citations:Zbl 0125.29601 PDFBibTeX XMLCite \textit{B. Valkó}, Acta Arith. 92, No. 4, 367--381 (2000; Zbl 0946.11006) Full Text: DOI EuDML