Distinguishing subgroups of the rationals by their Ramsey properties

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable.Comment: 14 page

Quasi-randomness and algorithmic regularity for graphs with general degree distributions

We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if $G=(V,E)$ satisfies a certain boundedness condition, then $G$ admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72–80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph $G$ in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without “dense spots.

A simple proof of distance bounds for Gaussian rough paths

We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their covariance functions. Similar estimates have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of our argument is that we obtain the bound for the third level iterated integrals merely based on the first two levels, and this reflects the intrinsic nature of rough paths. Our estimates are sharp when both covariance functions have finite 1-variation, which includes a large class of Gaussian processes. Two applications of our estimates are discussed. The first one gives the a.s. convergence rates for approximated solutions to rough differential equations driven by Gaussian processes. In the second example, we show how to recover the optimal time regularity for solutions of some rough SPDEs.Comment: 20 pages, updated abstract and introductio

Partition regularity with congruence conditions

An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist that each variable x_i is a multiple of some given d_i? This is a question of Hindman, Leader and Strauss. Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity.Comment: 5 page

Partition regularity of a system of De and Hindman

We prove that a certain matrix, which is not image partition regular over R near zero, is image partition regular over N. This answers a question of De and Hindman.Comment: 7 page

Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc