🤖 AI Summary
This work investigates the limitations of greedy strategies in constructing parity decision trees (PDTs), focusing on the existence of strongly folding directions within the Fourier spectrum support of Boolean functions. The authors present the first explicit class of Boolean functions devoid of ambient spectral components, for which the intersection of spectral supports under any two distinct translations is at most $O(|S|^{1/2})$, substantially tightening known upper bounds on spectral folding. By leveraging an affine subspace partitioning technique—APLPS—derived from full linear expansions, they achieve precise control over spectral support structure. Their results show that under the “lazy” maximal folding assumption, greedy methods cannot surpass the $O(|S|^{1/2})$ PDT depth upper bound, matching the best-known general bound and exposing the inadequacy of this assumption, though leaving open the potential advantages of adaptive greedy strategies.
📝 Abstract
We study Boolean functions and their Fourier spectrum supports in the context of parity decision trees (PDTs). Recently, H.~Hatami et al.~\cite{HHL+} constructed examples whose Fourier support \(\mathcal S\) satisfies $$ |(\mathcal S+γ_1)\cap(\mathcal S+γ_2)|=O(|\mathcal S|^{5/6}) $$ for all distinct \(γ_1,γ_2\), thereby refuting a natural greedy approach based on finding a single large folding direction. We strengthen this folding estimate by constructing an explicit infinite family of Boolean functions such that $$ |(\mathcal S+γ_1)\cap(\mathcal S+γ_2)|=O(|\mathcal S|^{1/2}) $$ for all distinct \(γ_1,γ_2\). The construction uses a special affine subspace partition, called an APLPS-partition, obtained from full linear spreads. In contrast with the probabilistic construction of \cite{HHL+}, our construction is explicit and has no background spectral components. We also discuss consequences for greedy approaches to PDT construction. Under the <<lazy>> assumption that the maximum-folding bound is inherited by all restrictions, the usual folding-counting argument cannot yield a PDT upper bound better than \(O(|\mathcal S|^{1/2})\), matching the known general upper bound. However, this inheritance assumption is false in general; hence our result refutes only this <<lazy>> maximum-folding approach, while a complete refutation of adaptive greedy strategies remains open.