A finitary version of Gromov's polynomial growth theorem

We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $R_0 > \exp(\exp(Cd^C))$ for which the number of elements in a ball of radius $R_0$ in a Cayley graph of $G$ is bounded by $R_0^d$, then $G$ has a finite index subgroup which is nilpotent (of step $<C^d$). An effective bound on the finite index is provided if "nilpotent" is replaced by 'polycyclic", thus yielding a non-trivial result for finite groups as well.Comment: 43 pages, no figures, to appear, GAFA. This is the final version; referee corrections have been incorporated, and some additional references added

Partition regularity and multiplicatively syndetic sets

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x, d, x + d, x + 2d\}$.Comment: 29 pages. v3. Referee comments incorporated, accepted for publication in Acta Arithmetic

A complex analogue of Toda's Theorem

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class \mathbf{P}^{#\mathbf{P}}, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #\mathbf{P}. This result, which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines \cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in \cite{BZ09} is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in \cite{BZ09} were semi-algebraic -- they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in \cite{BZ09} to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence we obtain a complex analogue of Toda's theorem. The results contained in this paper, taken together with those contained in \cite{BZ09}, illustrate the central role of the Poincar\'e polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers -- namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational Mathematic

Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem

Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.Comment: 14 pages, 6 figures, read at the Robert Clifton memorial conferenc

Regularity lemmas in a Banach space setting

Szemer\'edi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lov\'asz and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst, Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compact- ness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lov\'asz and Szegedy as well as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions, arXiv preprint arXiv:1401.2906 (2014)].Comment: 15 pages. The topological part has been substantially improved based on referees comments. To appear in European Journal of Combinatoric