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作者：Wu, Meng
摘要：We prove the following conjecture of Furstenberg (1969): if A, B subset of [0,1] are closed and invariant under xp mod 1 and xq mod 1, respectively, and if log p/log q is not an element of Q, then for all real numbers u and v, dim(H) (uA+ v) boolean AND B <= max {0, dim(H) A + dim(H) B - 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on R. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of plan...
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作者：Bordenave, Charles; Collins, Benoit
摘要：Let (sigma(1), . . . , sigma(d)) be a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n -> infinity, these permutations viewed as operators on the n-1 dimensional vector space {x(1), . . . , x(n)) is an element of C-n, Sigma x(i) = 0}, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refine...
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作者：Mok, Ngaiming; Pila, Jonathan; Tsimerman, Jacob
摘要：We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields. Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varietie...
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作者：Varju, Peter P.
摘要：The Bernoulli convolution nu(lambda) with parameter lambda is an element of (0, 1) is the probability measure supported on R that is the law of the random variable Sigma +/-lambda(n), where the +/- are independent fair coin-tosses. We prove that dim nu(lambda) = 1 for all transcendental lambda is an element of (1/2, 1).
