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On the p-adic Littlewood conjecture for quadratics

Friday 27th June 2014 - 14:30 to 15:30
INI Seminar Room 2
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 Teulié 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 Teullié proved that $lim inf_{q?1} q·log(q)·||qx||·|q|_p$ is finite and asked whether this limit is positive. We give a new proof of de Mathan and Teullié'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.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons