Multiplicatively badly approximable vectors

Vortrag von Dr. Reynold Fregoli

Datum: 03.10.22  Zeit: 13.30 - 14.30  Raum: Y27H28

The Littlewood Conjecture states that for all pairs of real numbers \((\alpha, \beta)\) the product \(\mid q\mid \mid q\alpha+p_1\mid\mid q\beta+p_2\mid\) becomes arbitrarily close to \(0\) when the vector \((q, p_1, p_2)\) ranges in \( \mathbb{Z}^3 \) and \( q \neq 0\). To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood conjecture, where the factor \(\mid q\mid\) is replaced by an increasing function \(f(\mid q\mid).\) More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions \(f\) for which the above product and its higher dimensional generalisations stay bounded away from \(0\) for at least one pair \((\alpha, \beta) \in\, \mathbb{R}^2.\) This problem happens to be intimately connected with the equidistribution rate of certain segments on the expanding torus in \( SL_3(\mathbb{R})/SL_3(\mathbb{Z})\) under the action of the full diagonal group.