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arXiv · 1712.02907

Equidistribution of dilated curves on nilmanifolds

Bryna Kra·Nimish Shah·Wenbo Sun
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Abstract

Generalizing classic results for a family of measures in the torus, for a family $(μ_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study conditions under which the family equidistributes, meaning conditions under which the measures $μ_t$ converge as $t\to\infty$ in the weak$^\ast$ topology to the Haar measure on $X$. We give general conditions on a family of measures defined by a dilation process, showing necessary and sufficient conditions for equidistribution as the family dilates, along with conditions such that this holds for all dilates outside some set of density zero. Furthermore, we show that these two types of equidistribution are different.

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Bryna Kra, Nimish Shah, Wenbo Sun. 2018-06-04. Equidistribution of dilated curves on nilmanifolds. https://doi.org/10.1112/jlms.12156

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