{"id":359840,"date":"2017-01-31T10:21:29","date_gmt":"2017-01-31T18:21:29","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=359840"},"modified":"2018-10-16T20:03:26","modified_gmt":"2018-10-17T03:03:26","slug":"absolute-continuity-bernoulli-convolutions-simple-proof","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/absolute-continuity-bernoulli-convolutions-simple-proof\/","title":{"rendered":"Absolute continuity of Bernoulli convolutions, a simple proof"},"content":{"rendered":"<p>The distribution <span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" style=\"display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.3333px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #404040; font-family: 'Times New Roman', Times, serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>&#x03BD;<\/mi><mi>&#x03BB;<\/mi><\/msub><\/math>\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"msubsup\"><span id=\"MathJax-Span-4\" class=\"mi\">\u03bd<\/span><span id=\"MathJax-Span-5\" class=\"mi\">\u03bb<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">\u03bd\u03bb<\/span><\/span> of the random series <span id=\"MathJax-Element-2-Frame\" class=\"MathJax\" style=\"display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.3333px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #404040; font-family: 'Times New Roman', Times, serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#x2211;<\/mo><mo>&#x00B1;<\/mo><msup><mi>&#x03BB;<\/mi><mi>n<\/mi><\/msup><\/math>\"><span id=\"MathJax-Span-6\" class=\"math\"><span id=\"MathJax-Span-7\" class=\"mrow\"><span id=\"MathJax-Span-8\" class=\"mo\">\u2211<\/span><span id=\"MathJax-Span-9\" class=\"mo\">\u00b1<\/span><span id=\"MathJax-Span-10\" class=\"msubsup\"><span id=\"MathJax-Span-11\" class=\"mi\">\u03bb<\/span><span id=\"MathJax-Span-12\" class=\"mi\">n<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">\u2211\u00b1\u03bbn<\/span><\/span> has been studied by many authors since the two seminal papers by Erd\\H{o}s in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba\\'{n}ski, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Garsia in 1962, that <span id=\"MathJax-Element-3-Frame\" class=\"MathJax\" style=\"display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.3333px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #404040; font-family: 'Times New Roman', Times, serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>&#x03BD;<\/mi><mi>&#x03BB;<\/mi><\/msub><\/math>\"><span id=\"MathJax-Span-13\" class=\"math\"><span id=\"MathJax-Span-14\" class=\"mrow\"><span id=\"MathJax-Span-15\" class=\"msubsup\"><span id=\"MathJax-Span-16\" class=\"mi\">\u03bd<\/span><span id=\"MathJax-Span-17\" class=\"mi\">\u03bb<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">\u03bd\u03bb<\/span><\/span> is absolutely continuous for a.e.\\ <span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" style=\"display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 17.3333px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #404040; font-family: 'Times New Roman', Times, serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>&#x03BB;<\/mi><mo>&#x2208;<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mrow class=\"MJX-TeXAtom-ORD\"><mo>\/<\/mo><\/mrow><mn>2<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math>\"><span id=\"MathJax-Span-18\" class=\"math\"><span id=\"MathJax-Span-19\" class=\"mrow\"><span id=\"MathJax-Span-20\" class=\"mi\">\u03bb<\/span><span id=\"MathJax-Span-21\" class=\"mo\">\u2208<\/span><span id=\"MathJax-Span-22\" class=\"mo\">(<\/span><span id=\"MathJax-Span-23\" class=\"mn\">1<\/span><span id=\"MathJax-Span-24\" class=\"texatom\"><span id=\"MathJax-Span-25\" class=\"mrow\"><span id=\"MathJax-Span-26\" class=\"mo\">\/<\/span><\/span><\/span><span id=\"MathJax-Span-27\" class=\"mn\">2<\/span><span id=\"MathJax-Span-28\" class=\"mo\">,<\/span><span id=\"MathJax-Span-29\" class=\"mn\">1<\/span><span id=\"MathJax-Span-30\" class=\"mo\">)<\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">\u03bb\u2208(1\/2,1)<\/span><\/span>. Here we give a considerably simplified proof of this theorem, using differentiation of measures instead of Fourier transform methods. This technique is better suited to analyze more general random power series.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The distribution \u03bd\u03bb\u03bd\u03bb of the random series \u2211\u00b1\u03bbn\u2211\u00b1\u03bbn has been studied by many authors since the two seminal papers by Erd\\H{o}s in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba\\'{n}ski, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Mathematical Research Letters","msr_number":"","msr_organization":"","msr_pages_string":"231-239","msr_page_range_start":"231","msr_page_range_end":"239","msr_series":"","msr_volume":"3","msr_copyright":"","msr_conference_name":"","msr_doi":"10.4310\/MRL.1996.v3.n2.a8","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"1996-01-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/dx.doi.org\/10.4310\/MRL.1996.v3.n2.a8","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-359840","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"1996-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"231-239","msr_chapter":"","msr_isbn":"","msr_journal":"Mathematical Research Letters","msr_volume":"3","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"359843","msr_publicationurl":"http:\/\/dx.doi.org\/10.4310\/MRL.1996.v3.n2.a8","msr_doi":"10.4310\/MRL.1996.v3.n2.a8","msr_publication_uploader":[{"type":"file","title":"MRL-1996-0003-0002-a008","viewUrl":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-content\/uploads\/2017\/01\/MRL-1996-0003-0002-a008.pdf","id":359843,"label_id":0},{"type":"url","title":"http:\/\/dx.doi.org\/10.4310\/MRL.1996.v3.n2.a8","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.4310\/MRL.1996.v3.n2.a8","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/dx.doi.org\/10.4310\/MRL.1996.v3.n2.a8"}],"msr-author-ordering":[{"type":"user_nicename","value":"peres","user_id":33234,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=peres"},{"type":"text","value":"Boris Solomyak","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"article","related_content":[],"_links":{"self":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/359840","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/359840\/revisions"}],"predecessor-version":[{"id":410264,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/359840\/revisions\/410264"}],"wp:attachment":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media?parent=359840"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=359840"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=359840"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=359840"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=359840"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=359840"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=359840"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=359840"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=359840"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=359840"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=359840"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=359840"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=359840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}