Algèbre et géométrie

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of […]

Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation x+z=2y has a solution in A.Finitary versions of Roth's theorem study subsets A of {0, ... , N}, and ask whether the same holds for sufficiently large N, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, […]