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Dynamics Seminar
February 17, 2009

Uri Shapira
Hebrew University

A solution to an open problem of Cassels on Diophantine properties of cubic numbers.

We prove existence of real numbers x,y, possessing the following property: For any real a,b, liminf |n| ||nx - a|| ||ny - b|| = 0, where ||c|| denotes the distance of c to the nearest integer. This answers a 50 years old question of Cassels. The most interesting part of the result is that there are algebraic numbers with the above property!

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