Absolute continuity of Bernoulli convolutions for algebraic parameters
By P. P. Varj'u. Journal of The American Mathematical Society, 2016
55 citations recorded by the source corpus.
<p>We prove that Bernoulli convolutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript lamda"> <mml:semantics> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are absolutely continuous provided the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an algebraic number sufficiently close to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depending on the Mahler measure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/cad6655bc02b413d9759d1141d38a187b40f933b
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In Applied Mathematics
In Geometry and Topology
- P. P. Varj'u
- Pablo Shmerkin
- Michael Hochman
- Balázs Bárány
- Ariel Rapaport
- Meng Wu
- Boris Solomyak
- Tuomas Orponen
In Mathematical Physics
- P. P. Varj'u
- Pablo Shmerkin
- Michael Hochman
- Ariel Rapaport
- Balázs Bárány
- Jonathan M. Fraser
- Meng Wu
- Boris Solomyak
Citations in the corpus
- On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “icate that Theorem 1.5 is optimal in a number of ways. Veryrecently,somestrikingprogressonthedimensionsandabsolutecontinuityof Bernoulli convolutions for algebraic parameters was achievedby P. Varju´ [49] and E. BreuillardandP.Varju´ [7]. Thelatterarticlealsouncoverssomedeepconnectionsbetween Bernou”
- Hausdorff dimension of planar self-affine sets and measures (2017, background): “ace the integrandlog(Q n(x)),whichisastepfunction,byacontinuousapproximation, usingapartitionofunitytoapproximatetheindicatorsofthelevelsets. Alternativelyonecanaveragetheentropyovertranslations,asin[44]. These “alternative” entropyfunctionscanbemadesothatthedifferencefromthe scale-nentropyisoforderO”
- On the dimension of Bernoulli convolutions for all transcendental parameters (2018, background): “e rich literature on Bernoulli convolutions. Instead, we refer the interested readers to the surveys [9,12,19,20]. The reader interested in Question 1 is also recommended to consult the recent papers [16,17,21]. Hochman [10]proved Theorem 3 conditionally on the hypothesis that there is a number C &g”
- Recent progress on Bernoulli convolutions (2016, background): “ractional derivatives in Lp for some p > 1 (depending on λ). Shmerkin and Solomyak [56], [57] extended these ideas to more general self-similar measures. We conclude with the following result from [61], which gives the first new examples of absolutely continuous Bernoulli convolutions since [23]. ”