# Fourier Dimension of Boolean functions

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?

• It's probably linear-invariant or even affine-invariant. May 12, 2022 at 5:20
• @YuvalFilmus would it be invariant under product or composition of $f$ with itself? May 14, 2022 at 7:26
• Why not check it using some examples? May 14, 2022 at 10:20