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YCor
YCor
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Equidistribution on subvarieties of $SL_n$\mathrm{SL}_n(\mathbb{Z})$

Let$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $SL_n(\mathbb{Z})$$\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates of coset representatives of $SL_n(\mathbb{Z})/\Gamma_{\infty}$$\SL_n(\mathbb{Z})/\Gamma_{\infty}$ are equidistributed in a suitable sense.

Do such generic equidistribution results persist when one replaces $SL_n(\mathbb{Z})$$\SL_n(\mathbb{Z})$ by a subset cut out from the $\mathbb{Z}$-integral points of a codimension 1 subvariety of $SL_n$$\SL_n$?

A concrete example: the Iwasawa $x$ coordinate of cosets of the form $SL_2(\mathbb{Z})/\Gamma_{\infty}$$\SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributes modulo $1$. Now, let $f(a,b,c,d) \in \mathcal{O}(SL_2)^{\Gamma_{\infty}}$$f(a,b,c,d) \in \mathcal{O}(\SL_2)^{\Gamma_{\infty}}$ be a polynomial (i.e. $f(a,b,c,d) = f(c,d)$). Is the Iwasawa $x$ coordinate of cosets of $SL_2(\mathbb{Z})/\Gamma_{\infty}$$\SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributed modulo $1$ as well?

Equidistribution on subvarieties of $SL_n(\mathbb{Z})$

Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates of coset representatives of $SL_n(\mathbb{Z})/\Gamma_{\infty}$ are equidistributed in a suitable sense.

Do such generic equidistribution results persist when one replaces $SL_n(\mathbb{Z})$ by a subset cut out from the $\mathbb{Z}$-integral points of a codimension 1 subvariety of $SL_n$?

A concrete example: the Iwasawa $x$ coordinate of cosets of the form $SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributes modulo $1$. Now, let $f(a,b,c,d) \in \mathcal{O}(SL_2)^{\Gamma_{\infty}}$ be a polynomial (i.e. $f(a,b,c,d) = f(c,d)$). Is the Iwasawa $x$ coordinate of cosets of $SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributed modulo $1$ as well?

Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$

$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates of coset representatives of $\SL_n(\mathbb{Z})/\Gamma_{\infty}$ are equidistributed in a suitable sense.

Do such generic equidistribution results persist when one replaces $\SL_n(\mathbb{Z})$ by a subset cut out from the $\mathbb{Z}$-integral points of a codimension 1 subvariety of $\SL_n$?

A concrete example: the Iwasawa $x$ coordinate of cosets of the form $\SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributes modulo $1$. Now, let $f(a,b,c,d) \in \mathcal{O}(\SL_2)^{\Gamma_{\infty}}$ be a polynomial (i.e. $f(a,b,c,d) = f(c,d)$). Is the Iwasawa $x$ coordinate of cosets of $\SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributed modulo $1$ as well?

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kindasorta
kindasorta
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Equidistribution on subvarieties of $SL_n(\mathbb{Z})$

Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates of coset representatives of $SL_n(\mathbb{Z})/\Gamma_{\infty}$ are equidistributed in a suitable sense.

Do such generic equidistribution results persist when one replaces $SL_n(\mathbb{Z})$ by a subset cut out from the $\mathbb{Z}$-integral points of a codimension 1 subvariety of $SL_n$?

A concrete example: the Iwasawa $x$ coordinate of cosets of the form $SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributes modulo $1$. Now, let $f(a,b,c,d) \in \mathcal{O}(SL_2)^{\Gamma_{\infty}}$ be a polynomial (i.e. $f(a,b,c,d) = f(c,d)$). Is the Iwasawa $x$ coordinate of cosets of $SL_2(\mathbb{Z})/\Gamma_{\infty}$ equidistributed modulo $1$ as well?