A metric theory of minimal gaps

We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)-\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence.Comment: Version 2: Fixed a small bug pointed out by Niclas Technau in the statement of section 3, and added references to few gaps in sequences of fractional parts suggested by Andrew Granvill

Observing Magnetic Fields on Large Scales

Observations of magnetic fields on scales up to several Mpc are important for understanding cluster and large-scale structure evolution. Our current census of such structures is heavily biased -- towards fields of several microG, towards fields in deep potential wells, and towards high inferred field strengths in cooling flow and other clusters from improper analysis of rotation measure data. After reviewing these biases, I show some recent results on two relics that are powered in very different ways. I describe new investigations that are now uncovering weak diffuse fields in the outskirts of clusters and other low density environments, and the good prospects for further progress.Comment: To be published in JKAS, from proceedings of the 3rd Korean Astrophysics Workshop, 16-20 Aug., 2004, "International Conference on Cosmic Rays and Magnetic Fields in Large Scale Structure." 7 pages, 7 color figure