פעילויות השבוע

In 2005, Bergelson, Host and Kra showed that if $(X,\mu,T)$ is an ergodic measure preserving system and $A\subset X$, then for every $\epsilon>0$ there exists a syndetic set of $n\in\mathbb{N}$ such that $\mu(A\cap T^{-n}A\cap\dots\cap T^{-kn}A)>\mu^{k+1}(A)-\epsilon$ for $k\leq3$, extending Khintchine‘s theorem. This phenomenon is called multiple recurrence with good lower bounds. Good lower bounds for certain polynomial expressions was studied by Frantzikinakis but several questions remain open. In this talk I will survey this topic, and present some progress regarding polynomial expressions, commuting transformations, and configurations involving the prime numbers. This is work in progress with Joel Moreira, Ahn Le and Wenbo Sun.

Experimentally, the convex-layer decomposition of subsets of the integer grid (”grid peeling“) seems to behave at the limit like the affine curve-shortening flow. We offer some theoretical arguments to explain this phenomenon. In particular, we derive some rigorous results for the special case of peeling the quarter-infinite grid: We prove that, in this case, the number of grid points removed up to iteration $n$ is $\Theta(n^{3/2}\log{ n})$ and moreover, the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor. Joint work with David Eppstein and Sariel Har-Peled

Measuring occurrence times of random events, aimed to determine the statistical properties of the governing stochastic process, is a basic topic in science and engineering, and has been the topic of numerous mathematical modeling techniques. Often, the true statistical properties of the random process deviate from the measured properties due to the so called ”dead time“ phenomenon, defined as a time period after a reaction in which the detection system is not operational. From a mathematical point of view, the dead time can be interpreted as a rarefied series of the original time series, obtained by removing all events which are within the dead time period inflicted by previous events.

When the waiting times between consecutive events form a series of independent identically distributed random variables, a natural setting for analyzing the distribution of the number of event- or the event counter- is a renewal process. In particular, for high rate measurements (or, equivalently, large measurement time), the limit distribution of the counter is well understood, and can be described directly through the first two moments of the waiting time between consecutive events.

In the talk we will discuss limit theorems for counters with paralyzing dead time (type II counter), expressed directly through the probability density function of the waiting time between consecutive events. This is done by writing explicit formulas for the for the first and second moments of a waiting time distribution between consecutive events in the rarefied process, in terms of the probability density function of the waiting of the original process.

Langlands Shahidi method is one of the two main approaches for defining and studying automorphic L-functions. This method is centered around Shahidi local coefficients which are analytic invariants associated with certain induced representations on linear groups. These coefficients arise from a uniqueness result known as uniqueness of Whittaker model. Among the local applications of these coefficients one finds irreducibility results and a formula for Plancherel measures. In the context of metaplectic groups, which are non-linear covering groups, uniqueness of Whittaker model does not hold anymore. Yet, an analog for these coefficients exists, dating back to Kazhdan-Patterson seminal work on the theta representations. In this talk we shall give new and simple interpretation to this analog for coverings of p-adic SL(2) and GL(2) and relate them to Tate gamma factors. We shall also give new formulas for the Plancherel measures and explain how to define gamma factors associated with covering groups using the local coefficients matrix.