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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)

##### MSC:
 11P21 Lattice points in specified regions 11B57 Farey sequences; the sequences $$1^k, 2^k, \dots$$
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