[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81906-en":3,"doc-seo-81906-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81906,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","On a Boolean Function Without Bold Folding in the Spectrum Support and Implications for Greedy Approaches to PDT Depth","Studies Boolean functions via the structure of their Fourier spectrum support in the setting of parity decision trees (PDTs). Strengthening earlier work, an explicit infinite family is constructed where for all distinct shifts the intersection size |(S+u1)∩(S+u2)| equals Θ(|S|^{1/2}). The construction uses a special affine subspace partition (APLP Spatition) derived from full linear spreads and avoids probabilistic spectral components. The results show that a common “lazy” maximum-folding inheritance argument cannot improve the PDT upper bound beyond Θ(|S|^{1/2}.), and refutes that limited greedy approach, while adaptive greedy strategies remain unresolved.","arXiv :2607 .04806v 1 [ cs .CC] 6 Jul 2026  \nOn a Boolean function without bold folding in the spectrum support and implications for greedy approaches to PDT depth  \nYuriy Tarannikov∗  \nAbstract  \nWe study Boolean functions and their Fourier spectrum supports in the context of parity decision trees (PDTs) . Recently, H. Hatami et al. [1] constructed examples whose Fourier support 􀁓 satisfies  \n| (􀁓 + 􀀍1) ∩ (􀁓 + 􀀍2)| = 􀁏 (|􀁓| 5/6)  \nfor all distinct 􀀍1 ,􀀍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  \n| (􀁓 + 􀀍1) ∩ (􀁓 + 􀀍2)| = 􀁏 (|􀁓| 1/2)  \nfor all distinct 􀀍1 ,􀀍2 . The construction uses a special affine subspace partition, called an APLPSpartition, obtained from full linear spreads. In contrast with the probabilistic construction of [1], 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 􀁏 (|􀁓| 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.  \nKey words: Boolean functions, spectrum support, folding, parity decision trees (PDT), log-rank conjecture, XOR functions, vector space partitions, linear spreads, greedy algorithms.  \nMSC 68Q11  \n1 Introduction  \nBoolean functions — mappings from F􀁮2 to {0, 1} or {−1, 1} — are fundamental objects in complexity theory, cryptography, and discrete mathematics. Their spectral analysis, based on the Fourier transform over the group F􀁮2, allows one to relate combinatorial properties of functions to algebraic characteristics such as matrix rank or spectrum support size. In this work we study the structure of the spectrum support of Boolean functions in the context of parity decision trees (PDTs) and their connection to the log-rank conjecture for XOR functions.  \nLet us clarify our notation. In cryptography and coding theory (e.g., [2, 3, 4]), Boolean functions are often viewed as mappings to {0, 1}, and the Walsh coefficients are defined as  \n􀁗􀁦 (􀁵) = ∑︁ (−1)􀁦 (􀁸)+⟨􀁵,􀁸⟩ ,  \n􀁸∈F􀁮2  \n∗ e-mail: [yutarann@gmail.com](yutarann@gmail.com)  \nwhere ⟨􀁵,􀁸⟩ = (︂􀁩1 􀁵􀁩 􀁸 􀁩 )︂ mod 2 .  \nIn complexity theory (in particular, [5, 1]), it is common to use functions taking values in ±1 and Fourier coefficients  \n(􀀋) = 21􀁮 􀁸􀁮2 􀁦 (􀁸)(−1)⟨􀀋,􀁸⟩ .  \nT(thehesesettwoo approachef nonzero cosearefficiequentsi)vaislentind:eifpe􀁧nd(􀁸e)nof( −1)nor􀁦m(􀁸a)li,stheatinon(􀀋In)t=he2−se􀁮q􀁗ue􀁦l (􀀋we) .fTheollowspthecetcrum supponventionorotf  \n[1], i.e. we consider functions with values in ±1 and use the normalised Fourier transform unless stated otherwise.  \nFor a Boolean function 􀁦 : F􀁮2 → {−1, 1}, we define its spectrum support  \n􀁓 = supp  = {􀀋 ∈ F􀁮2 | (􀀋)  0},  \nand denote 􀁫 = |􀁓| .  \nA central notion in our study is that of a folding in the spectrum support. Following [6], for 􀀍 ∈ F 􀁮2 we define the set  \n􀁏􀀍 = {︂ (􀀋,􀀌) ∈ (︂􀁓2)︂ | 􀀋 + 􀀌 = 􀀍 }︂ .  \nThe elements of 􀁏􀀍 are unordered pairs of distinct vectors from 􀁓 whose sum is 􀀍 . Such pairs are said to form a folding in direction 􀀍 . If |􀁏􀀍 | ≥ 2, then the direction 􀀍 contains at least two distinct pairs in the folding; if |􀁏􀀍 | is large, we speak of a «bold» folding.  \nIn [1] an equivalent but less transparent notation is used, namely intersections of shifts: for distinct 􀀍1 ,􀀍2 one considers  \n(􀁓 + 􀀍1 ) ∩ (􀁓 + 􀀍2 ) ,  \nwhere 􀁓 + 􀀍 = {􀀋 + 􀀍 | 􀀋 ∈ 􀁓} . It is easy to see that if 􀀍 = 􀀍1 + 􀀍2 , then  \n| (􀁓 + 􀀍1 ) ∩ (􀁓 + 􀀍2 ) | = 2|􀁏􀀍 | ,  \nsince each intersection gives a pair of endpoints corresponding to one pair in 􀁏 􀀍 . Thus the two descriptions are equivalent up to a ","cbCaijF0gzkIdWge","https://ap.wps.com/l/cbCaijF0gzkIdWge","pdf",510066,7,1,15,"English","en",105,"# Abstract\n# Introduction\n## Boolean functions and Fourier spectrum\n## Folding in spectrum support\n## Connection to communication complexity and log-rank conjecture\n## XOR functions and parity decision trees","[{\"question\":\"What is the main result about Fourier spectrum support intersections?\",\"answer\":\"The paper constructs an explicit infinite family of Boolean functions such that for any distinct shifts u1 and u2, the intersection |(S+u1) ∩ (S+u2)| equals Θ(|S|^{1/2}), improving on earlier Θ(|S|^{5/6}) counterexamples.\"},{\"question\":\"How does the construction differ from earlier probabilistic examples?\",\"answer\":\"Unlike the probabilistic construction, the paper’s family is explicit and has no background spectral components.\"},{\"question\":\"Why does the result matter for greedy approaches to PDT depth?\",\"answer\":\"Under the “lazy” assumption that a maximum-folding bound is inherited by all restrictions, the usual folding-counting argument cannot yield a PDT upper bound better than Θ(|S|^{1/2}), so the “lazy” maximum-folding greedy strategy is refuted, while fully adaptive greedy strategies remain open.\"}]","On a Boolean Function Without Bold Folding in the Spectrum Support and Implications for Greedy Approaches to PDT Depth | PDF",1784176981,38,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"on-a-boolean-function-without-bold-folding-in-the-spectrum-support-and-implications-for-greedy-approaches-to-pdt-depth","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/on-a-boolean-function-without-bold-folding-in-the-spectrum-support-and-implications-for-greedy-approaches-to-pdt-depth/81906/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-03","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What is the main result about Fourier spectrum support intersections?","Question",{"text":77,"@type":78},"The paper constructs an explicit infinite family of Boolean functions such that for any distinct shifts u1 and u2, the intersection |(S+u1) ∩ (S+u2)| equals Θ(|S|^{1/2}), improving on earlier Θ(|S|^{5/6}) counterexamples.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the construction differ from earlier probabilistic examples?",{"text":82,"@type":78},"Unlike the probabilistic construction, the paper’s family is explicit and has no background spectral components.",{"name":84,"@type":75,"acceptedAnswer":85},"Why does the result matter for greedy approaches to PDT depth?",{"text":86,"@type":78},"Under the “lazy” assumption that a maximum-folding bound is inherited by all restrictions, the usual folding-counting argument cannot yield a PDT upper bound better than Θ(|S|^{1/2}), so the “lazy” maximum-folding greedy strategy is refuted, while fully adaptive greedy strategies 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