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On irregularities of sums of integers. (English) Zbl 0946.11006

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

Citations:

Zbl 0125.29601
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