A uniform version of Jarník’s theorem. (English) Zbl 0924.11084

Plagne, Alain

Acta Arith. 87, No. 3, 255-267 (1999).

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)

MSC:

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

Keywords:

lattice points; Farey fractions; sequence of real numbers; convex curve