[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85888-en":3,"doc-seo-85888-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},85888,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Dense Subset Sum in Multi-Dimension","Dense subset sum in multi-dimension studies the additive structure of subset sums S(A) from a large set A of n vectors inside a d-dimensional box [N1]×…×[Nd]. The work characterizes when S(A) contains all integer points in large, well-structured regions and uses these structural properties to design efficient algorithms for deciding membership t∈S(A). Results focus on the dense regime n≫√Φ (Φ=∏Ni), improving prior multi-dimensional density thresholds and enlarging the guaranteed region.","arXiv :2607 . 10343v 1 [ cs .DS] 11 Jul 2026  \nDense Subset Sum in Multi-Dimension  \nLin Chen, Tingwei Hu, Yuchen Mao, Guochuan Zhang∗  \nJuly 14, 2026  \nAbstract  \nWe study the additive structure of dense subset sum in multi-dimension, and use the structure to develop efficient algorithms for the dense subset sum problem. More precisely, given a set A of n vectors in the d-dimensional hyperrectangle [N1] × [N2] × · · · × [Nd], we study the structure of S (A), which is the set of all subset sums of A, and then utilize the structure to develop efficient algorithms for determining whether t ∈ S (A) for a given t. We focus on the dense regime of the problem where n ≫ √Φ and Φ = N1 × · · · × Nd.  \nThe problem is well understood in the 1-dimensional case. It is known that if n ≫ √Φ , then S(A) contains an arithmetic progression of length Φ [Sárközy ’94, Szemerédi & Vu ’06] . If we further have that no prime can divide the majority of A, then S(A) contains all integers in the range [o(1)σ(A),(1 − o(1))σ(A)], where σ(A) = Pa∈A a [Lev ’03] . Using this combinatorics result, it has been shown that with the sole condition that n ≫ √Φ , then one can determine whBriethengmran∈&SW(Ae)llinnit’(n21)] . time for any t ∈ [o(1)σ(A),(1 − o(1))σ(A)] [Galil & Margalit ’91,  \nFor multi-dimension, the problem is far from clear. It is only known that if no non-trivial lattice (lattice other than Zd ) can contain the majority of A, then S (A) contains all integer points within some ellipsoid, provided that n ≫ Φ2/3 for d = 2 [Freiman’96, Plagne ’99], and n ≫ Φ ~~ d ~~−d~~1~~ for d ≥ 3 [Chaimovich ’91] .  \nWe improve upon these combinatorics results by showing that for any constant d ≥ 1, if n ≫ √Φ , then S (A) contains a long generalized progression in multi-dimension. If we further have that no non-trivial lattice can contain the majority of A, then S(A) contains all the integer points in the zonotope {x1 a 1 + · · · + xnan : o(1) ≤ xj ≤ 1 − o(1), xj ∈ R} . Compared to the previous results, our result significantly reduces the density threshold and enlarges the region inside which all the integer points belong to S (A) . Indeed, it matches the bound for the 1-dimensional case.  \nsumUspingroblour cem ionmbmiunalti-tdoirmicsenresusionlt, we also develop an (n)-time algorithm for the dense subset  \n∗ {linchen198662,tingweihu,maoyc,[zgc}@zju.edu.cn. Zhejiang University](zgc}@zju.edu.cn. Zhejiang University).  \nContents  \n1 Introduction 3  \n1. 1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.2 Other Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6  \n2 Preliminaries 7  \n2. 1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.2 Progressions and Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n2.3 Remainders Modulo A Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n2.4 Symmetry Sets and Kneser’s Theorem .......................... 10  \n2.5 Convex Analysis Tools ................................... 11  \n3 Long Progressions in Dense Subset Sums 12  \n3.1 Long Progressions with Independent Basis . . . . . . . . . . . . . . . . . . . . . . . . 12  \n3.2 Large Inradius of Zonotopes ................................ 15  \n3.3 All Lattice Points in a Large Hypercube ......................... 16  \n3.4 Long Progressions with Axis-Parallel Basis ....................... 19  \n4 A John-Type Theorem for Dense Subset Sum 19  \n5 An Algorithm for Dense Subset Sum 24  \n5.1 Computing Almost Common Lattice ........................... 24  \n5.2 Extracting Spread Subsets ................................. 27  \n5.3 Reducing to Modular Subset Sum ............................. 28  \n5.4 Putting Things Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  \n6 Extending to More General Cases 33  \n7 Generating Disjoint Subsets with Equal Sum 34  \n8 Generating Remainders 35  \n9 Compu","cbCaipaiXr5aL8Lt","https://ap.wps.com/l/cbCaipaiXr5aL8Lt","pdf",366805,2,1,51,"English","en",105,"# Contents\n## 1 Introduction\n## 2 Preliminaries\n## 3 Long Progressions in Dense Subset Sums\n## 4 A John-Type Theorem for Dense Subset Sum\n## 5 An Algorithm for Dense Subset Sum\n## 6 Extending to More General Cases\n## 7 Generating Disjoint Subsets with Equal Sum\n## 8 Generating Remainders\n## 9 Computing Modular Sumsets in Multi-dimension\n## Appendices","[{\"question\":\"What is the dense subset sum problem studied in the document?\",\"answer\":\"Given a set A of n vectors in a d-dimensional box, the document studies the subset-sum set S(A) and characterizes when a target t belongs to S(A). The focus is on the dense regime where n is much larger than √Φ, with Φ being the product of the side lengths.\"},{\"question\":\"What structural outcomes are proved about S(A) in multi-dimension?\",\"answer\":\"For any constant dimension d≥1, if n≫√Φ then S(A) contains a long generalized progression in multi-dimension. With additional non-lattice concentration conditions, S(A) also contains all integer points in a zonotope described by bounded coefficients along basis vectors.\"},{\"question\":\"How does the paper turn structure into computation?\",\"answer\":\"The document develops an algorithm for dense subset sum that uses lattice-related computations, extraction of spread subsets, reduction to modular subset sum, and recombination of results. The goal is to decide membership t∈S(A) efficiently using the established structural characterization.\"}]",1784206966,129,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"dense-subset-sum-in-multi-dimension","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/dense-subset-sum-in-multi-dimension/85888/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the dense subset sum problem studied in the document?","Question",{"text":75,"@type":76},"Given a set A of n vectors in a d-dimensional box, the document studies the subset-sum set S(A) and characterizes when a target t belongs to S(A). The focus is on the dense regime where n is much larger than √Φ, with Φ being the product of the side lengths.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What structural outcomes are proved about S(A) in multi-dimension?",{"text":80,"@type":76},"For any constant dimension d≥1, if n≫√Φ then S(A) contains a long generalized progression in multi-dimension. With additional non-lattice concentration conditions, S(A) also contains all integer points in a zonotope described by bounded coefficients along basis vectors.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper turn structure into computation?",{"text":84,"@type":76},"The document develops an algorithm for dense subset sum that uses lattice-related computations, extraction of spread subsets, reduction to modular subset sum, and recombination of results. 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