{"id":357491,"date":"2017-01-25T10:35:51","date_gmt":"2017-01-25T18:35:51","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357491"},"modified":"2018-10-16T19:59:40","modified_gmt":"2018-10-17T02:59:40","slug":"convolutions-cantor-measures-without-resonance","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/convolutions-cantor-measures-without-resonance\/","title":{"rendered":"Convolutions of Cantor Measures Without Resonance"},"content":{"rendered":"

Denote by \u03bc<\/span>a<\/span><\/span><\/span><\/span><\/span> the distribution of the random sum (<\/span>1<\/span>\u2212<\/span>a<\/span>)<\/span>\u2211<\/span>\u221e<\/span>j<\/span>=<\/span>0<\/span><\/span><\/span><\/span>\u03c9<\/span>j<\/span><\/span>a<\/span>j<\/span><\/span><\/span><\/span><\/span>, where P<\/span>(<\/span>\u03c9<\/span>j<\/span><\/span>=<\/span>0<\/span>)<\/span>=<\/span>P<\/span>(<\/span>\u03c9<\/span>j<\/span><\/span>=<\/span>1<\/span>)<\/span>=<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span> and all the choices are independent. For 0<\/span><<\/span>a<\/span><<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>, the measure \u03bc<\/span>a<\/span><\/span><\/span><\/span><\/span> is supported on C<\/span>a<\/span><\/span><\/span><\/span><\/span>, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (<\/span>1<\/span>\u2212<\/span>2<\/span>a<\/span>)<\/span><\/span><\/span><\/span>, and iterating this process inductively on each of the remaining intervals.
\nWe investigate the convolutions \u03bc<\/span>a<\/span><\/span>\u2217<\/span>(<\/span>\u03bc<\/span>b<\/span><\/span>\u2218<\/span>S<\/span>\u2212<\/span>1<\/span><\/span><\/span>\u03bb<\/span><\/span>)<\/span><\/span><\/span><\/span>, where S<\/span>\u03bb<\/span><\/span>(<\/span>x<\/span>)<\/span>=<\/span>\u03bb<\/span>x<\/span><\/span><\/span><\/span> is a rescaling map. We prove that if the ratio log<\/span><\/span>b<\/span>\/<\/span><\/span><\/span>log<\/span><\/span>a<\/span><\/span><\/span><\/span> is irrational and \u03bb<\/span>\u2260<\/span>0<\/span><\/span><\/span><\/span>, then<\/p>\n

D<\/span>(<\/span>\u03bc<\/span>a<\/span><\/span>\u2217<\/span>(<\/span>\u03bc<\/span>b<\/span><\/span>\u2218<\/span>S<\/span>\u2212<\/span>1<\/span><\/span><\/span>\u03bb<\/span><\/span>)<\/span>)<\/span>=<\/span>min<\/span>(<\/span>dim<\/span>H<\/span><\/span><\/span>(<\/span>C<\/span>a<\/span><\/span>)<\/span>+<\/span>dim<\/span>H<\/span><\/span><\/span>(<\/span>C<\/span>b<\/span><\/span>)<\/span>,<\/span>1<\/span>)<\/span>,<\/span><\/span><\/span><\/span><\/div>\n

where D<\/span><\/span><\/span><\/span> denotes any of correlation, Hausdorff or packing dimension of a measure.
\nWe also show that, perhaps surprisingly, for uncountably many values of \u03bb<\/span><\/span><\/span><\/span> the convolution \u03bc<\/span>1<\/span>\/<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u2217<\/span>(<\/span>\u03bc<\/span>1<\/span>\/<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u2218<\/span>S<\/span>\u2212<\/span>1<\/span><\/span><\/span>\u03bb<\/span><\/span>)<\/span><\/span><\/span><\/span> is a singular measure, although dim<\/span>H<\/span><\/span><\/span>(<\/span>C<\/span>1<\/span>\/<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>)<\/span>+<\/span>dim<\/span>H<\/span><\/span><\/span>(<\/span>C<\/span>1<\/span>\/<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>)<\/span>><\/span>1<\/span><\/span><\/span><\/span> and log<\/span><\/span>(<\/span>1<\/span>\/<\/span><\/span><\/span>3<\/span>)<\/span>\/<\/span><\/span><\/span>log<\/span><\/span>(<\/span>1<\/span>\/<\/span><\/span><\/span>4<\/span>)<\/span><\/span><\/span><\/span> is irrational.<\/p>\n","protected":false},"excerpt":{"rendered":"

Denote by \u03bca the distribution of the random sum (1\u2212a)\u2211\u221ej=0\u03c9jaj, where P(\u03c9j=0)=P(\u03c9j=1)=1\/2 and all the choices are independent. For 0<\/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":"Hebrew University Magnes Press","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Israel Journal of 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