Counting Diophantine approximations.(English)Zbl 1211.11085

Let positive real numbers $$\lambda_1,\ldots,\lambda_r$$ be given, such that $$1,\lambda_1,\ldots,\lambda_r$$ are linearly independent over $$\mathbb{Q}$$. Let $$Z_{\varepsilon}(N)$$ be the number of solutions to the simultaneous inequalities $|n_0\lambda_j-n_j|<\varepsilon\;\;\; (1\leq j\leq r)$ in square-free positive integers $$n_0,\ldots,n_r$$. Then it is shown that $Z_{\varepsilon}(N)=c\varepsilon^r N+o(N)\;\;\; (N\rightarrow\infty)$ for a suitable constant $$c>0$$, uniformly for $$0<\varepsilon\leq 1$$. This is just an example of a more general result in which the square-free numbers are replaced by an “extremal sequence” in the sense of the author’s work [Binary additive problems and the circle method, multiplicative sequences and convergent sieves. Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press, 91–132 (2009; Zbl 1193.11097)].
A key feature of the result is that it holds for all $$N$$, and not just for a lacunary sequence of values. Ideas of V. Bentkus and F. Götze [Ann. Math. (2) 150, No. 3, 977–1027 (1999; Zbl 0979.11048)] and D. E. Freeman [Mathematika 47, No. 1-2, 127–159 (2000; Zbl 1034.11027)], (which were developed in the context of the Davenport–Heilbronn circle method) are rephrased without recourse to exponential sums. The second major ingredient is then the author’s method of extremal sequences (loc. cit.). All that is needed to handle square-free numbers is the information that they form an extremal sequence.

MSC:

 11J13 Simultaneous homogeneous approximation, linear forms 11J25 Diophantine inequalities 11D75 Diophantine inequalities 11P55 Applications of the Hardy-Littlewood method
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