A solution to a problem of Cassels and Diophantine properties of cubic numbers

By Uri Shapira. 2008

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We prove that almost any pair of real numbers ; , satises the following inhomogeneous uniform version of Littlewood’s conjecture: 8; 2 R; lim inf jnj!1 jnjhn ihn i = 0; (C1) wherehi denotes the distance from the nearest integer. The existence of even a single pair that satises statement (C1), solves a problem of Cassels from the 50’s. We then prove that if 1;; span a totally real cubic number eld, then ; , satisfy (C1). This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfy Littlewood’s conjecture. It is further shown that if ; are any two real numbers, such that 1;; , are linearly dependent over Q, they cannot satisfy (C1). The results are then applied to give examples of irregular orbit closures of the diagonal group of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9bb4421e8d806e03546db6ac12a116522eee0c72

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Metric Diophantine Approximation : aspects of recent work (2016, background): “In a beautiful paper [88], Shapira establishes the following statement which solves a problem of Cassels dating back to the fifties.”