Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines. (English) Zbl 07834222
The setting considered here is the homogeneous space \(X={\mathrm{SL}}_3(\mathbb{R})/{\mathrm{SL}}_3(\mathbb{Z})\) and the action of \(g_t={\text{diag}}({\mathrm{e}}^{2t},{\mathrm{e}}^{-t},{\mathrm{e}}^{-t})\) on \(X\). Write \(\nu\) for the push-forward under the quotient map \({\mathrm{SL}}_3(\mathbb{R})\to X\) of a straight line segment in the expanding horosphere of \(\{g_t\mid t>0\}\). The main results here give necessary and sufficient Diophantine conditions on the line to ensure that various families of measures on \(X\) defined using the line equidistribute. These equidistribution results are applied to show that almost every point on the line \(y=ax+b\) is not Dirichlet-improvable if and only if one of the coefficients \(a,b\) is irrational.
Reviewer: Thomas B. Ward (Durham)
MSC:
| 37A17 | Homogeneous flows |
| 37A44 | Relations between ergodic theory and number theory |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 11J13 | Simultaneous homogeneous approximation, linear forms |
| 11J83 | Metric theory |
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