Mixing
If $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$, then $lim_{n rightarrow +infty} frac{|T_n Delta...
$begingroup$
Let ${T_n}_{n in mathbb{N}}$ be a Folner sequence and ${T'_n}_{n in mathbb{N}}$ be any sequences of subsets of $mathbb{Z}^d$ such that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$. Then, $lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = 0$, where $Delta$ means the symmetric difference.
${T_n}$ being a Folner sequence means that for any $v in mathbb{Z}^d$, $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, where $vT_n = {vt^{(n)} : t^{(n)} in T_n}$.
Using the fact that $A Delta C = (A Delta B) Delta (B Delta C)$, we can write, for any $v in mathbb{Z}^d$ and any $n in mathbb{N}$
begin{equation*}
T'_n Delta T_n = (T'_n Delta vT_n) Delta (vT_n Delta T_n).
end{equation*}
Then,
$$lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = lim_{n rightarrow +infty} frac{|(T'_n Delta vT_n) Delta (vT_n Delta T_n)|}{|T_n|} $$.
As ${T_n}_{n in mathbb{N}}$ is Folner, we know that $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, for any $v in mathbb{Z}^d$.
I guess I need to find a suitable $v$ such that $lim_{n rightarrow +infty} frac{|T'_n Delta vT_n|}{T_n} = 0$ and I guess the existence of this $v$ is guaranteed by the hypothesis that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$, but I am not sure if this is right and how I can justify that.
Could someone help me?
sequences-and-series group-theory
edited Jan 23 at 10:41
William Elliot
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
$endgroup$
add a comment |
$begingroup$
Let ${T_n}_{n in mathbb{N}}$ be a Folner sequence and ${T'_n}_{n in mathbb{N}}$ be any sequences of subsets of $mathbb{Z}^d$ such that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$. Then, $lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = 0$, where $Delta$ means the symmetric difference.
${T_n}$ being a Folner sequence means that for any $v in mathbb{Z}^d$, $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, where $vT_n = {vt^{(n)} : t^{(n)} in T_n}$.
Using the fact that $A Delta C = (A Delta B) Delta (B Delta C)$, we can write, for any $v in mathbb{Z}^d$ and any $n in mathbb{N}$
begin{equation*}
T'_n Delta T_n = (T'_n Delta vT_n) Delta (vT_n Delta T_n).
end{equation*}
Then,
$$lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = lim_{n rightarrow +infty} frac{|(T'_n Delta vT_n) Delta (vT_n Delta T_n)|}{|T_n|} $$.
As ${T_n}_{n in mathbb{N}}$ is Folner, we know that $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, for any $v in mathbb{Z}^d$.
I guess I need to find a suitable $v$ such that $lim_{n rightarrow +infty} frac{|T'_n Delta vT_n|}{T_n} = 0$ and I guess the existence of this $v$ is guaranteed by the hypothesis that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$, but I am not sure if this is right and how I can justify that.
Could someone help me?
sequences-and-series group-theory
edited Jan 23 at 10:41
William Elliot
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
$endgroup$
add a comment |
$begingroup$
Let ${T_n}_{n in mathbb{N}}$ be a Folner sequence and ${T'_n}_{n in mathbb{N}}$ be any sequences of subsets of $mathbb{Z}^d$ such that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$. Then, $lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = 0$, where $Delta$ means the symmetric difference.
${T_n}$ being a Folner sequence means that for any $v in mathbb{Z}^d$, $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, where $vT_n = {vt^{(n)} : t^{(n)} in T_n}$.
Using the fact that $A Delta C = (A Delta B) Delta (B Delta C)$, we can write, for any $v in mathbb{Z}^d$ and any $n in mathbb{N}$
begin{equation*}
T'_n Delta T_n = (T'_n Delta vT_n) Delta (vT_n Delta T_n).
end{equation*}
Then,
$$lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = lim_{n rightarrow +infty} frac{|(T'_n Delta vT_n) Delta (vT_n Delta T_n)|}{|T_n|} $$.
As ${T_n}_{n in mathbb{N}}$ is Folner, we know that $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, for any $v in mathbb{Z}^d$.
I guess I need to find a suitable $v$ such that $lim_{n rightarrow +infty} frac{|T'_n Delta vT_n|}{T_n} = 0$ and I guess the existence of this $v$ is guaranteed by the hypothesis that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$, but I am not sure if this is right and how I can justify that.
Could someone help me?
sequences-and-series group-theory
edited Jan 23 at 10:41
William Elliot
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
$endgroup$
Let ${T_n}_{n in mathbb{N}}$ be a Folner sequence and ${T'_n}_{n in mathbb{N}}$ be any sequences of subsets of $mathbb{Z}^d$ such that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$. Then, $lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = 0$, where $Delta$ means the symmetric difference.
${T_n}$ being a Folner sequence means that for any $v in mathbb{Z}^d$, $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, where $vT_n = {vt^{(n)} : t^{(n)} in T_n}$.
Using the fact that $A Delta C = (A Delta B) Delta (B Delta C)$, we can write, for any $v in mathbb{Z}^d$ and any $n in mathbb{N}$
begin{equation*}
T'_n Delta T_n = (T'_n Delta vT_n) Delta (vT_n Delta T_n).
end{equation*}
Then,
$$lim_{n rightarrow +infty} frac{|T_n Delta T'_n|}{|T_n|} = lim_{n rightarrow +infty} frac{|(T'_n Delta vT_n) Delta (vT_n Delta T_n)|}{|T_n|} $$.
As ${T_n}_{n in mathbb{N}}$ is Folner, we know that $lim_{n rightarrow +infty} frac{1}{|T_n|}|vT_n Delta T_n| = 0$, for any $v in mathbb{Z}^d$.
I guess I need to find a suitable $v$ such that $lim_{n rightarrow +infty} frac{|T'_n Delta vT_n|}{T_n} = 0$ and I guess the existence of this $v$ is guaranteed by the hypothesis that $lim_{n rightarrow +infty} frac{|T'_n|}{|T_n|} = 1$, but I am not sure if this is right and how I can justify that.
Could someone help me?
sequences-and-series group-theory
sequences-and-series group-theory
edited Jan 23 at 10:41
William Elliot
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
edited Jan 23 at 10:41
William Elliot
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
edited Jan 23 at 10:41
William Elliot
8,6672720
edited Jan 23 at 10:41
William Elliot
8,6672720
edited Jan 23 at 10:41
William Elliot
8,6672720
8,6672720
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
asked Jan 23 at 10:27
Luísa BorsatoLuísa Borsato
1,501315
1,501315
add a comment |
add a comment |
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