On a Boolean function without bold folding in the spectrum support and implications for greedy approaches to PDT depth

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 $$\mathrm{\mid }(\mathcal{S}+{\gamma }_1)\cap (\mathcal{S}+{\gamma }_2)\mathrm{\mid }=O(\mathrm{\mid }\mathcal{S}{\mathrm{\mid }}^{5\mathrm{/}6})$$ for all distinct ${\gamma }_1,{\gamma }_2$, 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 $$\mathrm{\mid }(\mathcal{S}+{\gamma }_1)\cap (\mathcal{S}+{\gamma }_2)\mathrm{\mid }=O(\mathrm{\mid }\mathcal{S}{\mathrm{\mid }}^{1\mathrm{/}2})$$ for all distinct ${\gamma }_1,{\gamma }_2$. 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 $O(\mathrm{\mid }\mathcal{S}{\mathrm{\mid }}^{1\mathrm{/}2})$, 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.