Common substring with shifts in b-ary expansions

Denote by S n ( x , y ) the length of the longest common substring of x and y with shifts in their first n digits of the b -ary expansions. We show that the sets of pairs ( x ,  y ), for which the growth rate of S n ( x , y ) is α log n with 0 ≤ α ≤ ∞ , have full Hausdorff dimension. Our method reli...

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Bibliographic Details
Published in:Archiv der Mathematik 2024-10, Vol.123 (4), p.369-377
Main Authors: Liao, Xin, Yu, Dingding
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Denote by S n ( x , y ) the length of the longest common substring of x and y with shifts in their first n digits of the b -ary expansions. We show that the sets of pairs ( x ,  y ), for which the growth rate of S n ( x , y ) is α log n with 0 ≤ α ≤ ∞ , have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices.
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-024-02038-1