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On the p-adic Littlewood conjecture for quadratics

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If you have a question about this talk, please contact Mustapha Amrani .

Interactions between Dynamics of Group Actions and Number Theory

Let |||| denote the distance to the nearest integer and, for a prime number p, let ||p denote the p-adic absolute value. In 2004, de Mathan and Teuli asked whether $inf{q?1} q||qx|||q|p = 0$ holds for every badly approximable real number x and every prime number p. When x is quadratic, the equality holds and moreover, de Mathan and Teulli proved that $lim inf{q?1} qlog(q)||qx|||q|_p$ is finite and asked whether this limit is positive. We give a new proof of de Mathan and Teulli’s result by exploring the continued fraction expansion of the multiplication of x by p with the help of a recent work of Aka and Shapira. We will also discuss the positivity of the limit.

This talk is part of the Isaac Newton Institute Seminar Series series.

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