We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such that $$\|g(j)-j\|\leq C_g,$$ for some constant $C_g$ which depends only on the element $g\in W(\mathbb{Z}^d)$. Consider the Hilbert space $L^2(\{0,1\}^{\mathbb{Z}^d},\mu)$, where $\{0,1\}^{\mathbb{Z}^d}$ comes with the standard Bernoulli measure $\mu$.
We are looking for a sequence of functions $f_n\in > L^2(\{0,1\}^{\mathbb{Z}^d},\mu)$ with $\|f_n\|_2=1$ and $$\|g.f_n-f_n\|_2\rightarrow 0, \text{ > for every } g\in W(\mathbb{Z}^d),$$ $$\|f_n\cdot\chi_{\{\omega_j\in > \{0,1\}^{\mathbb{Z}^d}:\omega_0=0\}}\|_2\rightarrow > 1.$$ where $\chi_{\{(\omega_j)_{j\in \mathbb{Z}^d}\in \{0,1\}^{\mathbb{Z}^d}:\omega_0=0\}}$ is the characteristic function of the cylinder set $\{(\omega_j)_{j\in \mathbb{Z}^d}\in \{0,1\}^{\mathbb{Z}^d}:\omega_0=0\}$
Motivation:
The existence of such a sequence for all $d$ would disprove the conjecture of Katok, that the interval exchange transformation group contains a free subgroup.
What we know about the question above:
In the joint work with Nicolas Monod, here, we showed that for $n=1$ the following function satisfy the properties above: $$f_n(\omega)=e^{-n \sum\limits_{j\in \mathbb{Z}} \omega_j e^{-\frac{|j|}{n}}}=\prod_{j\in Z} a_j^{\omega_j},$$ where $a_j=e^{-n e^{-\frac{|j|}{n}}}$. We are interested in extending this result to higher dimensions. Note that the function above is the product of functions of independent i.d. random variables.
Let $G< W(\mathbb{Z}^d)$ be a finitely generated subgroup of $W(\mathbb{Z}^d)$.
In addition to above (in collaboration with Nicolas Monod and Mikael de la Salle) we know that the existence of the functions with the property above in the class of functions which can be written as the product of functions of independent i.d. random variables is equivalent to a certain property of the Schreier graph of the action of $G$ on $\mathbb{Z}^d$. Let me give more details on this.
The Schreier graph of the action of $G$ on $X$ with respect to $S$ is the graph with vertices $X$ and with an edge between $x$ and $y$ for each $g \in S$ with $g x=y$.
We say that an infinite graph $G=(V,E)$ satisfies a Sobolev inequality rooted at $x_0 \in X$ if the value at $x_0$ of any $c_0$-function on $V$ is bounded by the $\ell^2$-norm of its gradient, i.e., there is a constant $C>0$ such that $$\|f\|_{c_0(V)} \leq C \sum_{x \sim x' \in V} |f(x') - f(x)|^2.$$
We can show that
The functions $f_n$ with the property above can be found in the class of products if and only if the Schreier graph of the action of $G$ on $X$ with respect to $S$ does not satisfy a Sobolev inequality.
Moreover, for $d=1,2$ and $G$ be a finitely generated subgroup of $W(n)$ with symmetric generating set $S$. Then the Schreier graph of the action of $G$ on $\mathbb{Z}^d$ does not satisfy a rooted Sobolev inequality. However there are subgroups in $W(\mathbb{Z}^3)$ such that their Schreier graph satisfies Sobolev inequality.
To summarize above, we can find a sequence of functions in the class of products with the above property only in cases $n=1,2$.
Any suggestions on potential examples of functions that can do the higher dimentional cases?
