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On the p-adic Littlewood conjecture for quadratics

Bengoechea, P (University of York)
Friday 27 June 2014, 14:30-15:30

Seminar Room 2, Newton Institute Gatehouse


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 Teuliť 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 Teulliť 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 Teulliť'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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