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On the distribution of the sequence \(n^2\theta\pmod 1\). (English) Zbl 0625.10029
About forty years ago, H. Heilbronn proved [Q. J. Math., Oxf. Ser. 19, 249-256 (1948; Zbl 0031.20502)], that for any \(\epsilon >0\) and any real \(\theta\) there exist infinitely many positive integers n such that \(\| \theta n^ 2\| \ll_{\epsilon}\quad n^{-+\epsilon},\) where \(\| \alpha \| =\min \{| \alpha -n|: n\in {\mathbb{Z}}\}.\) He suggested that the exponent \(-{1/2}+\epsilon\) may be replaced by \(- 1+\epsilon.\) The problem is tough and has resisted many attemps since then.
In the present paper the authors make the interesting conjecture: For any \(a,q\geq 1\) with \((a,q)=1\) we have \[ \sum_{1\leq m\leq M;\quad (m,q)=1}(m/q) e(ak^ 2\bar m/q) \ll_{\epsilon} (M^{1/2}+q^{- }M)(k^ 2,q)^{1/2} q^{\epsilon} \] where \((m/q)\) is the Jacobi symbol, \(\bar m\) is the multiplicative inverse of m mod q. Then they prove that, assuming the above conjecture, there exist infinitely many positive integers n such that \[ \| \theta n^ 2\| \ll_{\epsilon} n^{-(2/3)+\epsilon}. \]
Reviewer: Liu Ming-chit

MSC:
11J71 Distribution modulo one
11L40 Estimates on character sums
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