{"id":183019,"date":"2007-04-06T00:00:00","date_gmt":"2009-10-31T10:14:34","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/how-likely-is-buffons-needle-to-meet-a-cantor-square\/"},"modified":"2016-09-09T09:46:55","modified_gmt":"2016-09-09T16:46:55","slug":"how-likely-is-buffons-needle-to-meet-a-cantor-square","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/how-likely-is-buffons-needle-to-meet-a-cantor-square\/","title":{"rendered":"How likely is Buffon\u2019s needle to meet a Cantor square?"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let C<sub>n<\/sub> be the n\u2019th generation in the construction of the middle-half Cantor set. The Cartesian square K<sub>n<\/sub> of C<sub>n<\/sub> consists of 4<sup>n<\/sup> squares of side-length 1\/(4<sup>n<\/sup>). The chance that a long needle thrown at random in the unit square will meet K<sub>n<\/sub> is essentially the average length of the projections of K<sub>n<\/sub>. It is still an open problem to determine the exact rate of decay of this average.<\/p>\n<p>Until recently, the only explicit upper bound exp(- log_* n) was due to Peres and Solomyak.  (log_* n is the number of times one needs to take log to obtain a number less than 1, when starting from n). We obtain a much better bound by combining analytic and combinatorial ideas.<\/p>\n<p>This is joint work with Y. Peres and  A. Volberg.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let Cn be the n\u2019th generation in the construction of the middle-half Cantor set. The Cartesian square Kn of Cn consists of 4n squares of side-length 1\/(4n). The chance that a long needle thrown at random in the unit square will meet Kn is essentially the average length of the projections of Kn. It is [&hellip;]<\/p>\n","protected":false},"featured_media":194858,"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":[],"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-183019","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/cJp3jdEoY6c","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183019","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183019\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194858"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=183019"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=183019"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=183019"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=183019"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=183019"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=183019"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=183019"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=183019"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=183019"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=183019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}