A proof from my book. This theorem was needed to estimate (from below) the growth function of Okninski’s semigroup. For every natural number let denotes the number of primes . Say, , , , etc. The next theorem was proved by Chebyshev in 1850. We present a proof based on some ideas of Erdos (he […]

The first draft of a new paper “The Tarski numbers of groups” by Mikhail Ershov from Virginia, Gili Golan from Bar Ilan and myself can be found here: http://www.math.vanderbilt.edu/~msapir/egs/egs_1229.pdf. Here is an abstract: The Tarski number of a non-amenable group is the minimal number of pieces in a paradoxical decomposition of . In this paper […]

Here is a new paper of mine: http://front.math.ucdavis.edu/1310.7255. The story is this. A couple of years ago I was reading the paper http://arxiv.org/pdf/1010.1199.pdf (the first version). I was surprised finding there a statement answering a 30-years old question of van den Dries and Wilkie (if one asymptotic cone of a group is locally compact, should the group […]

I added a section about amenability (5.8), corrected many misprints: http://www.math.vanderbilt.edu/~msapir/book/b2.pdf .

Most probably I won’t be active on mathoverflow.net any longer: too many homework level questions and the community attitude towards these questions has become too liberal. I think I did help a few people with my answers, and I got some useful information from the answers to my questions.

Here is an adaptation of Lang’s proof of Gelfond’s theorem to prove that is not rational – needed for http://mathoverflow.net/questions/138247/prove-that-sqrt2-sqrt2-is-an-irrational-number-without-using-a-theorem/138291#138291. Consider the functions . Let , where is big enough (we shall see how large should be below). Suppose that is rational. Then is a rational number for every . Lemma 1. Let us have […]

Here are a few problems that I think are nice. Find a finitely presented group with uncountably many non-homeomorphic (or even non -equivalent) asymptotic cones, independent of Continuum Hypothesis. Recall that if we assume that CH is false than any uniform lattice in is an example (see Kramer, Shelah, Tent, Thomas, http://front.math.ucdavis.edu/0306.5420). The maximal number […]