VC-dimension–degree trade-offs over finite abelian groups

Establish whether every non-zero Boolean function \(f:G\to\{0,1\}\) on a finite abelian group \(G\cong\prod_{j=1}^r\mathbb{Z}_{m_j}\) satisfies \({\rm VC}(f)+\deg(f)\ge {\rm rank}(G)\), where \({\rm VC}(f)\) is the VC-dimension of the family of left-translates of the support of \(f\), \(\deg(f)\) is the Fourier degree based on the number of nonzero frequency coordinates, and \({\rm rank}(G)=r\).

Background

The paper defines Fourier degree for functions on a finite abelian group by expressing the function in the character basis and measuring the largest number of nonzero coordinates in a frequency vector supporting a nonzero Fourier coefficient.

For a subset $A\subseteq G$, the relevant VC-dimension is the VC-dimension of the family of its left-translates, following the notation introduced by Rodgers and Sahay. The established inequalities for the Boolean cube motivate asking whether a comparable uncertainty principle holds for arbitrary finite abelian groups.

The proposed lower bound uses the minimal number of cyclic factors in a decomposition of $G$ as the ambient group dimension and specializes to the paper’s Boolean-cube inequality when $G=\mathbb{Z}_2^n$.