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L. J. MORDELL 1. Let n > 2 be a given positive integer and (x) = (x1, x2, , x.) a given set of n integers. Let k be a given positive integer. Then L. Redei [1], improving an old result of Thue [2], has recently proved the THEOREM. If k is a prime, there exist integers a, b, (a, k) = 1, such that a set of integers (y), defined by yr-axr + b (mod k) (r t, 2, * I n), lies in an interval of width L where nLn-I _ 2n-1 kn-2 s.e. Iyr -Y I k, i.e. if nk-n < 2'-lkn-2 or k < 2n-1/n. There is also a trivial result L = 0 if we had allowed a= O (mod k). I notice that Redei's proof can be presented in a rather simpler form and also extended to the case where k is not a prime. We have now THEOREM I. Let 8 be the greatest divisor of k such that x1 =x2 = * = x. (mod 6), and so 6=(X1-Xn, X2-Xn, * * * , x.-ix, k). Then integers a 40 (mod k), b exist such that an integer set (y), defined by yr-axr+b(modk) (r= I * 2, lies in an interval of width L =L(6, k), where nL ? 2 n-la n-2 There is no loss of generality in theorems of this kind, if we suppose hereafter that 8 = 1. For clearly L(6, k) = 6L(1, k/I), since yr-ys a(xrx,) (mod k), and so we can put y, =8y' +c where Presented to the Society, September 5, 1953; received by the editors April 5, 1953. 854

On intervals containing an affinely equivalent set of 𝑛 integers mod 𝑘