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Run-length function of the Bolyai-Rényi expansion of real numbers

Rao LiFan LüLi Zhou — 2024

Czechoslovak Mathematical Journal

By iterating the Bolyai-Rényi transformation T ( x ) = ( x + 1 ) 2 ( mod 1 ) , almost every real number x [ 0 , 1 ) can be expanded as a continued radical expression x = - 1 + x 1 + x 2 + + x n + with digits x n { 0 , 1 , 2 } for all n . For any real number x [ 0 , 1 ) and digit i { 0 , 1 , 2 } , let r n ( x , i ) be the maximal length of consecutive i ’s in the first n digits of the Bolyai-Rényi expansion of x . We study the asymptotic behavior of the run-length function r n ( x , i ) . We prove that for any digit i { 0 , 1 , 2 } , the Lebesgue measure of the set D ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = 1 log θ i is 1 , where θ i = 1 + 4 i + 1 . We also obtain that the level set E α ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = α is of full Hausdorff dimension...

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