31,399 research outputs found

    Rational points near planar curves and Diophantine approximation

    Full text link
    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

    Get PDF
    In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation an=1x+1y\frac{a}{n}=\frac{1}{x}+\frac{1}{y}, in the case when aa is fixed and nn varies and in the case when both aa and nn vary.Comment: 16 pages, to appear in J. Number Theor
    • …
    corecore