Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in a set for its definition. As a special case, the spherical symmetrization of a indicator function is the indicator function on a ball centered at $0$ with same volume). Then (let me just state one case of results in Rogers' theorems 1 and 2 in the same paper. There are inequalities in higher moments as well.)
$$\int_X \left(\sum_{v \in \Lambda-0} f(v)\right)^2 d\mu(\Lambda) \le \int_X \left(\sum_{v \in \Lambda-0} f^*(v)\right)^2 d\mu(\Lambda) \tag{*}$$
where the integral is over the space of unimodular lattices equipped with the standard Haar measure.
But in Athreya and Marghulis' LOGARITHM LAWS FOR UNIPOTENT FLOWS paper, lemma 4.2, this is only stated as true for $n\ge 3$, and they dealt with $n=2$ separately without using Rogers' formula above.
So can anyone show me why (*) is not true for $n=2$? What should the correct statement for $n=2$ look like? Or is it just that the proof of Rogers' formula is wrong but the statement is still true?
A related thread for some mistake in Rogers' other paper for the case $n=2$: https://mathoverflow.net/a/282682/489992
