Journal of the
Korean Mathematical Society JKMS

We introduce the \textit{fractal expansions}, sequences of integers associated to a number. We show that these sequences characterize the $O$-sequences and encode some information about lex segment ideals. Moreover, we introduce numerical functions called \textit{fractal functions}, and we use them to solve the open problem of the classification of the Hilbert functions of any bigraded algebra.

Keywords: Hilbert function, lex segment ideal, bigraded algebra

MSC numbers: Primary 13F20, 13A15, 13D40