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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, March 14, 2024}
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\textbf{At} \emph{11:10 -- 12:00}
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\textbf{In} \emph{-101}

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{\large\scshape Ariel Rapaport 
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will talk about
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{\Large\bfseries Dimension of Bernoulli convolutions in R\^{}d\par}
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\textsc{Abstract:}
Let \$(\textbackslash{}lambda\_\{1\},\ldots{},\textbackslash{}lambda\_\{d\})=\textbackslash{}lambda\textbackslash{}in(0,1)\^{}\{d\}\$ be with \$\textbackslash{}lambda\_\{1\}\textgreater{}\ldots{}\textgreater{}\textbackslash{}lambda\_\{d\}\$ and let \$\textbackslash{}mu\_\{\textbackslash{}lambda\}\$ be the distribution of the random vector \$\textbackslash{}sum\_\{n\textbackslash{}ge0\}\textbackslash{}pm (\textbackslash{}lambda\_\{1\}\^{}\{n\},\ldots{},\textbackslash{}lambda\_\{d\}\^{}\{n\})\$, where the \$\textbackslash{}pm\$ are independent fair coin-tosses. Assuming \$P(\textbackslash{}lambda\_\{j\})\textbackslash{}ne 0\$ for all \$1\textbackslash{}le j\textbackslash{}le d\$ and nonzero polynomials with coefficients \$\textbackslash{}pm1,0\$, we show that \$\textbackslash{}operatorname\{dim\}\textbackslash{}mu\_\{\textbackslash{}lambda\}=\textbackslash{}min \textbackslash{}big(d,\textbackslash{}dim\_\{L\}\textbackslash{}mu\_\{\textbackslash{}lambda\} \textbackslash{}big)\$, where \$\textbackslash{}dim\_\{L\}\textbackslash{}mu\_\{\textbackslash{}lambda\}\$ is the Lyapunov dimension. This extends to higher dimensions a result of Varjú from 2018 regarding the dimension of Bernoulli convolutions on the real line. Joint work with Haojie Ren.








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