Rational group ring elements with kernels having irrational dimension

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.Comment: 35 pages, 1 figure; [TDA 3/12/13:] Updated version incorporating referee's suggestion

Quantitative equidistribution for certain quadruples in quasi-random groups

In a recent paper (arXiv:1211.6372), Bergelson and Tao proved that if $G$ is a $D$-quasi-random group, and $x$,$g$ are drawn uniformly and independently from $G$, then the quadruple $(g,x,gx,xg)$ is roughly equidistributed in the subset of $G^4$ defined by the constraint that the last two coordinates lie in the same conjugacy class. Their proof gives only a qualitative version of this result. The present notes gives a rather more elementary proof which improves this to an explicit polynomial bound in $D^{-1}$.Comment: 5 pages; [TDA Jun 6, 2014] Updated with reference to arxiv:1405.5629 [v3:] This preprint has been re-written to correct to a mistake in the proof of Corollary 3. The journal published that correction in a separate erratu