Abstract
Some number-theoretic problems involve the number of points with integer coordinates very near some smooth plane curves. This chapter introduces the basic results and some refinements of the theory. Some criteria are investigated and the theorem of Huxley and Sargos is studied in detail. In the section Further Developments, we prove a particular case of a general theorem given by Filaseta and Trifonov improving on the distribution of squarefree numbers in short segments. The dual problem of square-full numbers in short intervals is also investigated.
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- 1.
Some authors also use the vocable lattice points.
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For instance, the authors did not make use of Gorny’s inequality but proved a Landau–Hadamard–Kolmogorov like result similar to (5.20) using divided differences.
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Bordellès, O. (2012). Integer Points Close to Smooth Curves. In: Arithmetic Tales. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4096-2_5
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