Effective Ratner theorem for \(\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2\) and gaps in \(\sqrt{n}\;\mathrm{modulo}\,1\) (Q2822139)

\textit{N. D. Elkies} and \textit{C. T. McMullen} [Duke Math. J. 123, No. 1, 95--139 (2004; Zbl 1063.11020)] linked the unusual distribution of gaps in the sequence \(\{\sqrt{n} \text{ mod } 1\}\) (first noticed by numerical experiments in [\textit{M. D. Boshernitzan}, J. Anal. Math. 62, 225--240 (1994; Zbl 0804.11046)]) to the equidistribution of a one-parameter unipotent flow on the space of two-dimensional affine unimodular lattices \(\mathrm{SL}(2, \mathbb R) \ltimes \mathbb R^2/ \mathrm{SL}(2, \mathbb Z) \ltimes \mathbb Z^2\). They posed the question of whether this equidistribution could be made effective. Via exponential sum estimates, the authors effectivize this equidistribution. They observe that there are possible optimizations for this effectifization.