# Siegel's Mean Value Theorem by Rogers and Macbeath

I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that $$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ 0 \neq u \in \Lambda} f(v) d \mu(\Lambda)= \int_{\mathbb{R}^n}f(x) dx$$ where $\mu$ is the normed Haar measure on $SL(n, \mathbb{R})$.

However, I believe that I've found some inconsistencies in the work of Rogers and Macbeath. I would be very grateful if someone who is familiar with this field could confirm these, or point out that I'm wrong.

• In [1] I strongly suspect theorem 1, pg. 147 to be wrong. The theorem is a formula for the integral containing a sum over $l$ linear independent primitive points. In the case $l=2$ and after normalising the Haar measure in the theorem states $$\int_F \sum_{...} f(u,v)d \mu(\Lambda) = \frac{1}{\zeta(n)\zeta(n-1)} \int_{\mathbb{R}^{n^2}} f(x,y) d(x,y)$$ where the sum on the left ranges over all primitive points $u,v \in \Lambda$ with $\dim(u,v)=2$. $F$ is a fundamental domain for $SL(n,\mathbb{R})/SL(n,\mathbb{Z})$ here. However, this contradicts other results, e.g. by Rogers himself where the factor on the right hand side should be $\frac{1}{\zeta(n)^2}$ instead.
• Also, in my opinion in [1] lemma 9, pg.150, should be wrong as well, because the desired representation should not be unique.
• In [2], pg. 252, Rogers emphasises that theorem 5 works for $n \geq 2$. Although, in the proof of this theorem Rogers uses theorem 3 of the same paper which should not apply in the case $n=2$. This really surprised me, because Rogers explicitly highlights that the theorem 5 does apply for $n=2$ in the text above theorem 5.

[1] Siegel's mean value theorem in the geometry of numbers
A. M. Macbeath, C. A. Rogers http://www.ams.org/mathscinet-getitem?mr=0103183

[2] Mean values over the space of lattices C. A. Rogers http://www.ams.org/mathscinet-getitem?mr=0075243

• Where did these papers appear? – paul garrett Feb 8 '14 at 23:14
• Unfortunately, I couldn't find a free access via google(I looked beforehand and would had link both). If you have access to a mathscinet subscription you can find them here: ams.org/mathscinet-getitem?mr=0103183 and ams.org/mathscinet-getitem?mr=0075243 – Soeren Feb 8 '14 at 23:28
• Ah, thanks. Even if something's behind a paywall, just telling what journal (if any) it appeared in helps specify it... – paul garrett Feb 8 '14 at 23:30
• Yes, you are absolutely right! I added both links to my original post. – Soeren Feb 9 '14 at 2:09
• In Theorem 1 of paper [1], Rogers makes a comment about some sets being measure zero in $SL(n,\mathbf{R})/SL(n, \mathbf{Z})$. I tried but couldn't prove it even for $n \ge 3$. Does anybody know how to do it? – Mahbub Alam Jan 23 '18 at 11:07

In this field it is a well known fact that Rogers had an error in his paper for the $n=2$ case. It was corrected by Schmidt in his paper A Metrical Theorem in the Geometry of numbers