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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)N−1∕2+𝜖 where ∥α∥ denotes 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)∥ < CN−1∕g(R)+𝜖 for i = 1,⋯,R, where g(1) = 3 and
g(R) = 4g(R − 1) + 4R + 2 for R ≥ 2.
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