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 A numerical property of Hilbert functions and lex segment ideals J. Korean Math. Soc.Published online March 6, 2020 Giuseppe Favacchio Università degli Studi di Catania Abstract : 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 a 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 : 13F20, 13A15, 13D40 Full-Text :