[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86358-en":3,"doc-seo-86358-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86358,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","Algorithmic Information Bounds for Distances and Orthogonal Projections","Introduces a new technique for establishing lower bounds on the Kolmogorov complexity of geometric objects in Euclidean space, including points and lines. Applies the method to two results in algorithmic information theory with consequences for geometric measure theory. For independent points x and y in the plane, the distance |x−y| preserves at least half the complexity of x, yielding an improved Hausdorff-dimension bound for pinned distance sets. Proves a parallel statement for orthogonal projections, generalizing Bourgain’s exceptional set theorem.","arXiv :2509 .052 1 1v2 [ cs .CC] 11 Jul 2026  \nALGORITHMIC INFORMATION BOUNDS FOR DISTANCES AND  \nORTHOGONAL PROJECTIONS  \nPETER CHOLAK, MARIANNA CS¨ORNYEI, NEIL LUTZ, PATRICK LUTZ, ELVIRA MAYORDOMO,  \nAND DONALD M. STULL  \nAbstract. We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point x in the plane and anyother point y sufficiently independent of x, the distance between x and y retains at least half the complexity of the original point x. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer’s distance set conjecture. Second, we prove an analogous result for orthogonal projections:  \nfor any point x in the plane and any line through the origin which is sufficiently independent of x, the projection of x onto that line retains at least half the complexity of x. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.  \n1. Introduction  \nThis paper is a contribution to a growing area of research which connects algorithmic information theory and geometric measure theory and which is based on a close correspondence between Kolmogorov complexity and Hausdorff dimension. This correspondence has two facets. First, thereis a conceptual analogy between Kolmogorov complexity and Hausdorff dimension. For example, using Kolmogorov complexity, one can define a notion of effective dimension for objects such as infinite binary sequences and real numbers which shares many features with Hausdorff dimension [4, 16, 39] . Second, there is a specific technical result, proved by J. Lutz and N. Lutz and known asthe point-to-set principle, which allows one to use facts about Kolmogorov complexity and effective dimension to prove theorems about Hausdorff dimension [18] .  \nIn recent years, the point-to-set principle has been applied to make progress on a few long-standing questions in geometric measure theory. For example, N. Lutz and Stull used it to improve the best then-known lower bound for the Furstenberg set conjecture (a generalization of the Kakeya conjecture) [25] and Stull used it to improve the best known lower bound for Falconer’s distance set conjecture [41], an important open question in geometric measure theory [13] .  \nThe main goal of this paper is to introduce a new technique for proving lower bounds on the Kolmogorov complexity of geometric objects in Euclidean space. One of our primary motivations for introducing this new technique is that, when used in conjunction with the point-to-set principle, it seems especially useful for proving results in geometric measure theory which had previously appeared difficult to approach using the methods of algorithmic information theory. However, even ignoring these applications to geometric measure theory, we believe that the theorems in algorithmic information theory which can be proved using our technique are interesting in their own right.  \nIn this paper, we will give two example applications of our technique. In each case, we will first use our technique to prove a theorem on algorithmic information theory and then use the point-to-set principle to transfer this theorem to the setting of geometric measure theory. We have included these applications not just for their own sake, but also as a demonstration of the utility of our technique, which we believe likely has further applications within geometric measure theory. Infact, in unpublished work, Fiedler [8] has already used our technique to prove a new theorem on  \n2 P. CHOLAK, M. CS¨ORNYEI, N. LUTZ, P. LUTZ, E. MAYORDOMO, AND D. STULL  \nthe Hausdorff dimension of planar","cbCaiuyqrPlSNn19","https://ap.wps.com/l/cbCaiuyqrPlSNn19","pdf",410444,6,1,22,"English","en",105,"# Introduction\n## Overview and motivation\n## Point-to-set principle and prior applications\n## Main goals and technique\n## Application 1: distances and pinned distance sets\n## Application 2: orthogonal projections","[{\"question\":\"What new technique does the paper introduce?\",\"answer\":\"The paper introduces a technique for proving lower bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines.\"},{\"question\":\"How does the paper relate point independence to distance complexity?\",\"answer\":\"For a point x in the plane and another point y sufficiently independent of x, the distance |x−y| retains at least half the complexity of x, up to an o(r) term in the formal theorem statement.\"},{\"question\":\"What is the orthogonal-projection result and its consequence?\",\"answer\":\"For a point x and a line through the origin sufficiently independent of x, the orthogonal projection of x onto that line retains at least half the complexity of x, which leads to a generalization of Bourgain’s theorem on exceptional sets for orthogonal projections.\"}]",1784210828,55,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"algorithmic-information-bounds-for-distances-and-orthogonal-projections","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/algorithmic-information-bounds-for-distances-and-orthogonal-projections/86358/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What new technique does the paper introduce?","Question",{"text":76,"@type":77},"The paper introduces a technique for proving lower bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper relate point independence to distance complexity?",{"text":81,"@type":77},"For a point x in the plane and another point y sufficiently independent of x, the distance |x−y| retains at least half the complexity of x, up to an o(r) term in the formal theorem statement.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the orthogonal-projection result and its consequence?",{"text":85,"@type":77},"For a point x and a line through the origin sufficiently independent of x, the orthogonal projection of x onto that line retains at least half the complexity of x, which leads to a generalization of Bourgain’s theorem on exceptional sets for orthogonal 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