When structures are almost surely connected.
A generalization of Cobham's theorem for regular sequences.
Partition identities. I: Sandwich theorems and logical 0-1 laws.
Counting rooted trees: the universal law .
Characteristic points of recursive systems.
Asymptotics for the probability of connectedness and the distribution of number of components.
Cobham’s theorem and its extensions
Cobham’s theorem says that if and are two multiplicatively independent integers and is a - and -automatic sequence, then is eventually periodic. We give a summary of recent work on automatic sequences and their relation to Cobham’s theorem.
Logarithmic frequency in morphic sequences
We study the logarithmic frequency of letters and words in morphic sequences and show that this frequency must always exist, answering a question of Allouche and Shallit.
3x+1 inverse orbit generating functions almost always have natural boundaries
The 3x+k function sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of . We consider the generating functions , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...
Diagonalization and rationalization of algebraic Laurent series
We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime the reduction modulo of the diagonal of a multivariate algebraic power series with integer coefficients is an algebraic power series of degree at most and height at most , where is an effective constant that only depends on...
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