{"id":1115,"date":"2024-09-14T13:55:29","date_gmt":"2024-09-14T20:55:29","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1115"},"modified":"2024-09-16T19:40:25","modified_gmt":"2024-09-17T02:40:25","slug":"recent-progress-on-fourier-decay-of-probability-measures","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2024\/09\/14\/recent-progress-on-fourier-decay-of-probability-measures\/","title":{"rendered":"Recent progress on Fourier decay of probability measures"},"content":{"rendered":"<p>The HADES seminar on Tuesday, <strong>September 17th<\/strong>, will be at <strong>2<\/strong><strong>:00pm<\/strong>\u00a0in\u00a0<strong>Room 740.<\/strong><\/p>\n<p><strong>Speaker:<\/strong> <a href=\"https:\/\/math.berkeley.edu\/~ztao\/\">Zhongkai Tao<\/a><\/p>\n<p><strong>Abstract: <\/strong>Let <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712m_5750148431135043817l0.5542478677405105\" class=\"CToWUd\" title=\"\\mu\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_Na56r_H0PEWk091EE2myV9aA_nJPZTD77U-1uu9D1N1zvKnOscr3LFny3QfBQ2x-cug4oww41t1wRVjw3DsKMK8d_T5xE46cErDg7Mb83hrF8RQqnggiUw21tWj=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%5Cmu\" alt=\"\\mu\" width=\"8\" height=\"11\" data-bit=\"iit\" \/> be a probability measure on <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712m_5750148431135043817l0.007289447553399109\" class=\"CToWUd\" title=\"\\mathbb{R}^n\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NarulH_FS9Ql9kW9zAQ1_rVOPJCX86bRhrwV6u25hnQKz0AgTyiJ5_Q9dJssP65VjeFj6FD1iYydlVm3t4V4_3qkLYHkCSMe4vzrCmLpKEDQVIwQP1zdLeyVwjy-vMBkc5CeKCWMMlpTzqv=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%5Cmathbb%7BR%7D%5En\" alt=\"\\mathbb{R}^n\" width=\"18\" height=\"11\" data-bit=\"iit\" \/>. The Fourier transform of the measure, defined by <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712m_5750148431135043817l0.8515514534693134\" class=\"CToWUd\" title=\" \\hat{\\mu}(\\xi) = \\int e^{i x \\cdot \\xi } d\\mu(x)\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NbxPTtfQ1UQoj45LsTi-iVmXQovawoo-emteiUAoySjP9jf6N2MHKMYQP07IvJJ9sqamfbk0bXwoTI59PYFELVJGIgBSsf3cM0RBih76bMz7-ob6AH77yJj0kkutk0WXxLyueL0KDFAB8EjN-u1CeOJ6BzAB2qSQ9yWzOZ8ALAzk9O4LGRud524VfHpm5Tk78l2mf0jH9gHexnNnuBRmKi60GPX7aBXrTj2WFIhTeFEPxXvSAxiu8049tiQps4V52XT3OwY=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%5Cdisplaystyle%09%09%5Chat%7B%5Cmu%7D(%5Cxi)%09=%09%5Cint%09e%5E%7Bi%09x%09%5Ccdot%09%5Cxi%09%7D%09d%5Cmu(x)\" alt=\" \\hat{\\mu}(\\xi) = \\int e^{i x \\cdot \\xi } d\\mu(x)\" width=\"137\" height=\"36\" data-bit=\"iit\" \/> has been very useful in dynamical systems. A central question is the Fourier decay, that is, the uniform decay rate of <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712m_5750148431135043817l0.874621990961999\" class=\"CToWUd\" title=\"\\hat{\\mu}(\\xi)\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NanfQDsg5vtX94hvFivW3K5pUGaCb8oUkUOwWQErlg3-eqFnuYykTtOkrdM0ENsDh0mS_Ajb0iDj5hmhUu7Yt1vQ_Fbl-fMzpiemR9k_e8CeAMHeKsMKaHbcPfq68Kheu2SjGi60Nv6lrnJwPBzYw=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%5Chat%7B%5Cmu%7D(%5Cxi)\" alt=\"\\hat{\\mu}(\\xi)\" width=\"28\" height=\"16\" data-bit=\"iit\" \/>. This was studied by Erd\u0151s and Salem almost a\u00a0century ago. While polynomial Fourier decay, i.e. <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712m_5750148431135043817l0.5778865320173785\" class=\"CToWUd\" title=\"|\\hat{\\mu}(\\xi)| \\leq C|\\xi|^{-\\beta}\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NbFrTfq9QHPpfP7bW5rC0JoOIsSdyeSpRS0BgPukOMJTESL46lCEbKUY-OLattjRngpuszCV0Y2Pumc09jzBG7l0kWPcG4J_vDY6CSlpk041YCFEKAczxAgmvm9VVBT9DZHAfNq6nLkXMz7oA2elcv1mPNc_34Q6tECk-FmpSSlGiLMeJ60sATxT426rhH8X53U_X-c5wfVwL4Wg--R2A=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7C%5Chat%7B%5Cmu%7D(%5Cxi)%7C%09%5Cleq%09C%7C%5Cxi%7C%5E%7B-%5Cbeta%7D\" alt=\"|\\hat{\\mu}(\\xi)| \\leq C|\\xi|^{-\\beta}\" width=\"105\" height=\"18\" data-bit=\"iit\" \/>\u00a0for some <img loading=\"lazy\" decoding=\"async\" id=\"m_-8600708017411837712l0.633885397856425\" class=\"CToWUd\" title=\"\\beta&gt;0\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NahC82pF2rW0EJ2SmQFv7JnEJeMHjwGSubIJciF1Y8eHwDPaJ-DPT3XUiiLzIOTFlu-TD-01SFvinyvWGUpzzCcgrF7i8H-MPBXo_ohJFDFboCh5bjP_jKIZ5_4rXHrCIss=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%5Cbeta%3E0\" alt=\"\\beta&gt;0\" width=\"39\" height=\"14\" data-bit=\"iit\" \/>,\u00a0is expected in many situations, it is only recently that people can prove polynomial\u00a0Fourier decay for nontrivial measures coming from dynamical systems, c.f. the works of Bourgain, Dyatlov, Li, Sahlsten, Shmerkin, Orponen, de Saxc\u00e9, Khalil, Baker, Algom, Rodriguez Hertz, Wang, etc. I will try to explain the key ideas in some recent developments: sum-product estimates, additive combinatorics and the use of dynamical systems.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, September 17th, will be at 2:00pm\u00a0in\u00a0Room 740. Speaker: Zhongkai Tao Abstract: Let be a probability measure on . The Fourier transform of the measure, defined by has been very useful in dynamical systems. A central question is the Fourier decay, that is, the uniform decay rate of . This was [&hellip;]<\/p>\n","protected":false},"author":92,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[],"class_list":["post-1115","post","type-post","status-publish","format-standard","hentry","category-fall-2024"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1115","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/92"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=1115"}],"version-history":[{"count":3,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1115\/revisions"}],"predecessor-version":[{"id":1120,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1115\/revisions\/1120"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}