Vol. 45, No. 1, 1973

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On the fractional parts of a set of points. II

Roger Cook

Vol. 45 (1973), No. 1, 81–85
Abstract

Heilbronn proved that for any 𝜖 > 0 there exists a number C(𝜖) such that for any real numbers 𝜃 and N 1 there is an integer n such that 1 n N and n2𝜃< C(𝜖)N12+𝜖 where αdenotes the difference between α and the nearest integer, taken positively. The method depends on Weyl’s estimates for trigonometric sums. The result was generalized by Davenport who obtained analogous results for polynomials which have no constant term.

The object here is to obtain a result for simultaneous approximations to quadratic polynomials f1,fR having no constant term: For any 𝜖 > 0 there is a number C = C(𝜖,R) such that for any N 1 there is an integer n such that 1 n N and fi(n)< CN1∕g(R)+𝜖 for i = 1,,R, where g(1) = 3 and g(R) = 4g(R 1) + 4R + 2 for R 2.

Mathematical Subject Classification
Primary: 10F40
Milestones
Received: 14 December 1971
Published: 1 March 1973
Authors
Roger Cook