Published ETD Collection
Permanent URI for this collectionhttps://hdl.handle.net/10657/2
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Browsing Published ETD Collection by Subject "$p$-adic solenoids"
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Item Topics in Diophantine Approximation(2022-05-08) Chen, Huayang; Haynes, Alan; Koivusalo, Henna; Ott, William; Török, AndrewWe 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 $\delta$. As an application, we prove that the expected value of the smallest denominator is asymptotic, as $\delta\rar 0$, to $(8\sqrt{2}/\pi^2)\delta^{-1/2}.$