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LSpice
LSpice
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First of all, $Y$ is not called the ``grid space''“grid space”. It is sometimes called the affine space and can be identified with a quotient of the affine group $ASL_{n}$$\operatorname{ASL}_{n}$, namely the semi-direct product of $SL_{n}$$\operatorname{SL}_{n}$ with $R^{n}$$\mathbb R^{n}$.

Anyhow, the question you are referring to was addressed by U. Shapira in his well-known thesis where he studied the ``big“big Littlewood conjecture''conjecture”.

The main new ingredient here (for which one needs to assume $n\geq 3$, compared to the results of Einsiedler-Katok-LindenstraussEinsiedler–Katok–Lindenstrauss over the Littlewood conjecutre) is the famous $\times 2, \times 3$$\times 2$, $\times 3$ theorem of Furstenberg, showing that for any irrational number $\alpha$, the set $\left\{2^{n}3^{m}\alpha\right\}$ is dense modulo $1$. This theorem has been generalized by D. Berend, a student of Fursternberg's, to actions of commuting matrices over $\mathbb{T}^d$. Using Berend's theorem (with the observation that the derived $A$-action over the fiber (so one has a higher-rank commutative Anosov action over the torus)), your question follows (you can also consider the PhD thesis of Z. Wang).

Notice that in order to conclude your question, you just need to show that the origin of your fiber is a point contained in the orbit closure. On the other hand, due to Berened's theorem, if the origin is not a point in your orbit closure, the orbit will be stuck in some rational subgroup, so it is essentially if and only if.

First of all, $Y$ is not called the ``grid space''. It is sometimes called the affine space and can be identified with a quotient of the affine group $ASL_{n}$, namely the semi-direct product of $SL_{n}$ with $R^{n}$.

Anyhow, the question you are referring to was addressed by U. Shapira in his well-known thesis where he studied the ``big Littlewood conjecture''.

The main new ingredient here (for which one needs to assume $n\geq 3$, compared to the results of Einsiedler-Katok-Lindenstrauss over the Littlewood conjecutre) is the famous $\times 2, \times 3$ theorem of Furstenberg, showing that for any irrational number $\alpha$, the set $\left\{2^{n}3^{m}\alpha\right\}$ is dense modulo $1$. This theorem has been generalized by D. Berend, a student of Fursternberg's to actions of commuting matrices over $\mathbb{T}^d$. Using Berend's theorem (with the observation that the derived $A$-action over the fiber (so one has a higher-rank commutative Anosov action over the torus), your question follows (you can also consider the PhD thesis of Z. Wang).

Notice that in order to conclude your question, you just need to show that the origin of your fiber is a point contained in the orbit closure. On the other hand, due to Berened's theorem, if the origin is not a point in your orbit closure, the orbit will be stuck in some rational subgroup, so it is essentially if and only if.

First of all, $Y$ is not called the “grid space”. It is sometimes called the affine space and can be identified with a quotient of the affine group $\operatorname{ASL}_{n}$, namely the semi-direct product of $\operatorname{SL}_{n}$ with $\mathbb R^{n}$.

Anyhow, the question you are referring to was addressed by U. Shapira in his well-known thesis where he studied the “big Littlewood conjecture”.

The main new ingredient here (for which one needs to assume $n\geq 3$, compared to the results of Einsiedler–Katok–Lindenstrauss over the Littlewood conjecutre) is the famous $\times 2$, $\times 3$ theorem of Furstenberg, showing that for any irrational number $\alpha$, the set $\left\{2^{n}3^{m}\alpha\right\}$ is dense modulo $1$. This theorem has been generalized by D. Berend, a student of Fursternberg's, to actions of commuting matrices over $\mathbb{T}^d$. Using Berend's theorem (with the observation that the derived $A$-action over the fiber (so one has a higher-rank commutative Anosov action over the torus)), your question follows (you can also consider the PhD thesis of Z. Wang).

Notice that in order to conclude your question, you just need to show that the origin of your fiber is a point contained in the orbit closure. On the other hand, due to Berened's theorem, if the origin is not a point in your orbit closure, the orbit will be stuck in some rational subgroup, so it is essentially if and only if.

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Asaf
Asaf
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First of all, $Y$ is not called the ``grid space''. It is sometimes called the affine space and can be identified with a quotient of the affine group $ASL_{n}$, namely the semi-direct product of $SL_{n}$ with $R^{n}$.

Anyhow, the question you are referring to was addressed by U. Shapira in his well-known thesis where he studied the ``big Littlewood conjecture''.

The main new ingredient here (for which one needs to assume $n\geq 3$, compared to the results of Einsiedler-Katok-Lindenstrauss over the Littlewood conjecutre) is the famous $\times 2, \times 3$ theorem of Furstenberg, showing that for any irrational number $\alpha$, the set $\left\{2^{n}3^{m}\alpha\right\}$ is dense modulo $1$. This theorem has been generalized by D. Berend, a student of Fursternberg's to actions of commuting matrices over $\mathbb{T}^d$. Using Berend's theorem (with the observation that the derived $A$-action over the fiber (so one has a higher-rank commutative Anosov action over the torus), your question follows (you can also consider the PhD thesis of Z. Wang).

Notice that in order to conclude your question, you just need to show that the origin of your fiber is a point contained in the orbit closure. On the other hand, due to Berened's theorem, if the origin is not a point in your orbit closure, the orbit will be stuck in some rational subgroup, so it is essentially if and only if.