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][1] 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.)

$$\int_X \left(\sum_{v \in \Lambda-0} f(v)\right)^2 d\mu(\Lambda) \le \int_X \left(\sum_{v \in \Lambda-0} f^*(v)\right)^2 d\mu(\Lambda) \tag{*}$$

But in [Athreya and Marghulis' LOGARITHM LAWS FOR UNIPOTENT FLOWS paper, lemma 4.2][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 $n=2$? Or is it just that the proof of Rogers' formula is wrong but the statement is still true?


  [1]: https://academic.oup.com/plms/article/s3-6/2/305/1487949
  [2]: https://arxiv.org/pdf/0811.2806.pdf#page=9#