I was recently reading about Fourier dimension of Boolean functions. What I understand is that if we take the Fourier expansion of $f: \{\pm1\}^n \to \{\pm1\}$ and consider the monomials with non zero Fourier coefficients as vectors in $\mathbb{F}_2^n$, then the Fourier dimension is the dimension of the span of these vectors.
This might help to understand when a Boolean function is "reduceable" or "equivalent" to another one. For example, if we replace a variable with a product of say $m$ variables, then in the vectors of the corresponding monomials, those $m$ places will be $1$ or $0$ together. This gives us that the Fourier dimension remains unchanged in such a transformation.
My question is for what kind of variable transformations does the Fourier dimension remain unchanged?
