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A uniform version of Jarník’s theorem. (English) Zbl 0924.11084
Nearly 75 years ago V. Jarník proved the following result: Let \(\mathbb{Z}\) be the set of integers and \(q_0\) any positive integer. Then there exists a strictly convex curve \(C_0\) in \([0,1]^2\) and an integer \(q\geq q_0\) such that \[ | C_0\cap (\tfrac 1q \mathbb{Z})^2| \asymp q^{2/3}. \] In this paper the author shows: Let \((r_n)\) be any sequence of real numbers. Then there exists a strictly convex curve \(C\) and a strictly increasing sequence of integers \((q_n)\) with \(q_n\geq r_n\) such that \[ | C\cap (\tfrac 1q \mathbb{Z})^2| \gg \frac{q_n^{2/3}} {K^n}, \] where \(K\) is a constant and has the value \(K=15,\dots\;\).
Reviewer: E.Krätzel (Wien)

11P21 Lattice points in specified regions
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
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