Fix a prime power $q$. A triangle in $\\mathbb{F}_q^n$ is a triple $x,y,z$ of elements with $x+y=z$. Green’s triangle removal lemma in $\\mathbb{F}_q^n$ states that for each $\\epsilon>0$ there is a $\\delta>0$ such that every subset of $\\mathbb{F}_q^n$ which has few triangles (less than $\\delta N^2$, where $N=q^n$ is the total number of elements), can be made triangle-free by removing few elements (less than $\\epsilon N$). Previously, the best known bound was of tower type. Using a result of Kleinberg-Speyer, which builds on the breakthrough work of Croot-Lev-Pach on the cap set problem, and subsequent work by Ellenberg-Gijswijt and Blasiak-Church-Cohn-Grochow-Umans, we prove a polynomial bound for Green’s triangle removal lemma, which we can prove is tight for small q, and conjecture it is tight for all q. Joint work with Jacob Fox.<\/p>\n","protected":false},"excerpt":{"rendered":"
Fix a prime power $q$. A triangle in $\\mathbb{F}_q^n$ is a triple $x,y,z$ of elements with $x+y=z$. Green’s triangle removal lemma in $\\mathbb{F}_q^n$ states that for each $\\epsilon>0$ there is a $\\delta>0$ such that every subset of $\\mathbb{F}_q^n$ which has few triangles (less than $\\delta N^2$, where $N=q^n$ is the total number of elements), can […]<\/p>\n","protected":false},"featured_media":308285,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[13546],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-253664","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/i1z2aEegvYM","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/253664","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/253664\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media\/308285"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=253664"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=253664"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=253664"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=253664"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=253664"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=253664"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=253664"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=253664"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=253664"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=253664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}