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The fractional dimensional theory in Lüroth expansion

Luming ShenKui Fang — 2011

Czechoslovak Mathematical Journal

It is well known that every x ( 0 , 1 ] can be expanded to an infinite Lüroth series in the form of x = 1 d 1 ( x ) + + 1 d 1 ( x ) ( d 1 ( x ) - 1 ) d n - 1 ( x ) ( d n - 1 ( x ) - 1 ) d n ( x ) + , where d n ( x ) 2 for all n 1 . In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets F φ = { x ( 0 , 1 ] : d n ( x ) φ ( n ) , n 1 } are completely determined, where φ is an integer-valued function defined on , and φ ( n ) as n .

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