31,399 research outputs found
Rational points near planar curves and Diophantine approximation
In this paper, we establish asymptotic formulae with optimal errors for the
number of rational points that are close to a planar curve, which unify and
extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani.
Furthermore, we complete the Lebesgue theory of Diophantine approximation on
weakly non-degenerate planar curves that was initially developed by
Beresnevich-Zorin in the divergence case.Comment: 27 pages, corrected typos, to appear in Adv. Mat
Mean Value Theorems for Binary Egyptian Fractions
In this paper, we establish two mean value theorems for the number of
solutions of the Diophantine equation , in
the case when is fixed and varies and in the case when both and
vary.Comment: 16 pages, to appear in J. Number Theor
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