Topics in Diophantine Approximation

Date

2022-05-08

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Abstract

We discuss some topics about number theory including continued fractions, Hausdorff measure, p-adic analysis and analytic number theory in the preliminary knowledge part. In the next section, we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and the Duffin-Schaeffer theorem. We prove a complementary result, showing that the set of badly approximable points has maximum Hausdorff dimension. Our proof is a simple application of the elegant machinery of Schmidt's game. Moreover, we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius δ. As an application, we prove that the expected value of the smallest denominator is asymptotic, as δ\rar0, to (82/π2)δ−1/2.

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Keywords

$p$-adic solenoids, Badly approximable points, Schmidt's game, Hausdorff dimension, Smallest denominator, Probability mass function, Mobius inversion formula, Euler's summation formula, Expected value

Citation

Portions of this document appear in: Chen, A. Haynes: Badly approximable points for diagonal approximation in solenoids, Acta Arith. 199 (2021), no. 2, 153–161.