Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of |f(A,B)| for finite subsets A, B of a field, and a polynomial f(x,y) of the form f(x,y) = g(x) + yh(x), where the degree of g is greater than that of h.
We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation...
Page 1