This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.
A. Katok and F. Rodriguez Hertz established the following related result in @cite_43 which they call an arithmeticity theorem .
@cite_43
2041977310 2042993834 2609098951 2964092005
We show how interval arithmetic can be used in connection with Borsuk's theorem to computationally prove the existence of a solution of a system of nonlinear equations. It turns out that this new test, which can be checked computationally in several different ways, is more general than an existing test based on Miranda's theorem in the sense that it is successful for a larger set of situations. A numerical example is included. The following theorem is proved. Let f(x_1, , x_m) be a binary nonzero polynomial of m variables of degree . H the number of binary m -tuples (a_1, , a_m) with f(a_1, , a_m) = 1 is less than 2^ m- +1 , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. where m + and 3 . This theorem completely characterizes the codewords of the th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (x n ) there exists a subsequence x d , x 2d , x 3d ,…, x kd , for some positive integers k and d, such that ( i=1 ^k x_ id >C ). The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C = 1 a human proof of the conjecture exists; for C = 2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C = 3.