{"id":3354,"date":"2024-01-26T15:01:20","date_gmt":"2024-01-26T23:01:20","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=3354"},"modified":"2024-02-20T17:41:38","modified_gmt":"2024-02-21T01:41:38","slug":"antc-seminar-pete-clark-university-of-georgia","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-pete-clark-university-of-georgia\/","title":{"rendered":"The restricted variable Kakeya problem (Pete Clark, University of Georgia)"},"content":{"rendered":"<div>For a finite field F_q, a subset of F_q^N is a\u00a0<b>Kakeya set<\/b> if it contains a line in every direction (i.e., a coset of every one-dimensional linear subspace).\u00a0 The finite field Kakeya problem is to determine the minimal size K(N,q) of a Kakeya set in F_q^N.\u00a0 This problem was posed by Wolff in 1999 as an analogue to the Kakeya problem in Euclidean N-space, which was (and still is) one of the major open problems in harmonic analysis.\u00a0 It caused quite a stir in 2008 when Zeev Dvir showed that for each fixed N, as q -&gt; oo, K(N,q) is bounded below by a constant times q^N: the Euclidean analogue of this result is not only proved but known to be false.<\/div>\n<div><\/div>\n<div>But what about the constant?\u00a0 In 2009 Dvir-Kopparty-Saraf-Sudan gave a lower bound on K(N,q) that was within a factor of 2 of an upper bound due to Dvir-Thas.\u00a0 (I will briefly mention recent work of Bukh-Chao giving a decisive further improvement, but that is not the focus of the talk.) The key to this improved lower bound is<b>\u00a0<\/b>a\u00a0<b>multiplicity enhancement<\/b> of a 1922 result of Ore. In this talk I want to give my own exposition of this work together with a mild generalization: if X is a subset of F_q^N \\ {0}, then an <b>X-Kakeya set\u00a0<\/b>is a subset that contains a translate of the line generated by x for all x in X.\u00a0 Putting K_X(N,q) to be the minimal size of an X-Kakeya set in F_q^N, I will give a lower bound on K_X(N,q) that recovers the DKSS bound when X = F_q^N \\ {0}.\u00a0 This is similar in spirit to\u00a0 &#8220;statistical Kakeya&#8221; results of Dvir and DKSS but not overlapping much; in fact, I will give a statistical generalization of my result as well.<\/div>\n","protected":false},"excerpt":{"rendered":"<p>For a finite field F_q, a subset of F_q^N is a\u00a0Kakeya set if it contains a line in every direction (i.e., a coset of every one-dimensional linear subspace).\u00a0 The finite [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-3354","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-antc-seminar","cat_antc-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - 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