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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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