Approximate groups and doubling metrics (Q2883234)

This is an interesting, and (like all other attempts) only partially successful approach to find the analog of Freiman's theorem in noncommutative groups. Let \(G\) be a hereditarily monomial group (this assumption makes it possible to use some Fourier analysis) and \(A\) a finite symmetric normal set such that \(|A^n | \leq n^d |A| \) for all \(n\). (In commutative groups, an assumption for \(n=2\) suffices.) Then \(A\subset B\) with an \(O(d \log^3 d)\) dimensional ball \(B\) in a translation-invariant pseudometric \(\rho\) such that \( |B| \leq \exp \bigl( O( d \log d) \bigr) |A|\). The \textit{dimension} of a ball is defined as a number \(d\) such that \( |B(\rho, 2\delta)| \leq 2^d |B(\rho, \delta)| \) for sufficiently small \(\delta\).NEWLINENEWLINE NEWLINEThe main difference from other variants is that the covering structure is typically defined synthetically as a properly generalized arithmetic progression, and the structure of the balls as defined here needs to be clarified.NEWLINENEWLINE NEWLINEFor nilpotent groups the assumption can be weakened to \( |A|^3 \leq K |A| \). (A bound on \( |A^2| \) is generally insufficient.)NEWLINENEWLINE\{The article under review changed its title from `From polynomial growth to metric balls in monomial groups' to the present one. See \url{http://arxiv.org/abs/0912.0305}.\}