This will be a talk for the Mathematics Colloquium at Vassar College, November 10, 2015, tea at 4:00 pm, talk at 4:15 pm, Rockefeller Hall 310
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Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum introduced just recently, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.
This talk is based in part on current joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.
My Lecture Notes are available.
I found a paper on internet entitled “Riemann Rearrangement Theorem for some types of convergence “(http://homepages.math.uic.edu/~kslutsky/papers/2n-sum-range.pdf). The authors investigate the RRT for different types of convergence. I think maybe this paper is interesting for you. In addition there is a question: How much your theorem depends on the nature of usual convergence on real line? More precisely can one derive your theorem from some assumptions on convergence type? Clearly RRT is one of the needed assumption!! I am waiting to see your published paper.
Thanks for the comment and link! (For some reason, your link didn’t work, but this one did: http://homepages.math.uic.edu/~kslutsky/papers/2n-sum-range.pdf, even though they look the same to me at first glance.)
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