The topological property of the irregular sets on the lengths of basic intervals in beta-expansions.

Let β > 1 be a real number. A basic interval of order n is a set of real numbers in ( 0 , 1 ] having the same first n digits in their β -expansion which contains x ∈ ( 0 , 1 ] , denote by I n ( x ) and write the length of I n ( x ) as | I n ( x ) | . In this paper, we prove that the extremely irregular set containing points x ∈ [ 0 , 1 ] whose upper limit of − log β ⁡ | I n ( x ) | n equals to 1 + λ ( β ) is residual for every λ ( β ) > 0 , where λ ( β ) is a constant depending on β . [ABSTRACT FROM AUTHOR]