Programa de Doctorado en Investigación Matemática, UCM. Coorganized by Prof. Jesús Llorente Jorge, Department of Mathematical Analysis and Applied Mathematics, UCM).
In the dates February 18, 20, 25, and 27, 2025; March 4, 6, 11, 13, 18, and 20, 2025,
Professor Franciszek Prus-Wiśniowski
Instytut Matematyki, Uniwersytet Szczeciński, Poland
will deliver the Research Minicourse entitled
SCHEDULE: The course will take place on Tuesdays and Thursdays from 4:00 PM to 5:30 PM for 5 weeks. The first session will be on Tuesday, February 18, 2025, and the last session will be on Thursday, March 20, 2025.
LOCATION: Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid.
ABSTRACT.
Achievement sets (that is, sets of all subsums of a sequence or a series) are deeply rooted in real analysis, but they keep growing beyond the classic theory interconnecting with other topics like dynamical systems, fractal theory or number theory. The investigation of the topological nature of sets of all subsums of an absolutely convergent real series began over one century ago with a paper Soichi Kakeya [1] (see also [2]). The complete topological classification of achievement sets on the real line was established by J.A. Guthrie and J.E. Nymann in 1988 [4]. Achievement sets served as a counterexample to the Palis hypothesis from 1987 (see [3]) which stated that the arithmetic difference of two Cantor sets, both with Lebesgue measure zero, is either of Lebesgue measure zero or has non-empty interior. The investigations of achievement sets gained momentum about ten years ago, mainly in Poland and Ukraine, and nowadays the first monography on the topic is being written.
During the 15 hour lecture, the core facts on achievement sets will be presented. Among others, we will cover the Guthrie-Nymann Classification Theorem, the special role of multigeometric series, the quite new and very useful metric invariant which is the center of distances, and also algebraic operations on achievement sets. Some topics, like the cardinal functions or the multidimensional achievement sets, will not fit in the short lecture, unfortunately. A number of still open problems, some very elementary in formulation, will be presented with the hope that some of those attending the lecture will get caught by their simplicity and beauty, and will get involved in search for the answers.
PREREQUISITES:
Standard knowledge of calculus (basics of real series) and of general topology (elementary concepts and rudimentary metric spaces). Audience is also assumed to be familiar with the concept of Lebesgue measure on the real line. Very occasionally, some more advanced tools will be mentioned or used (like iterated function systems or Hausdorff dimension), but they are not fundamental for the core of this lecture. The lecture will be given in English.
STRUCTURE:
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Week 1 (February 18 and 20): Preliminaries: a classification of real series, perfect sets, linearly ordered sets, Cantor sets, Cantorvals.
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Week 2 (February 25 and 27): The Guthrie-Nymann classification: subsums of absolutely convergent series, four gap lemmas, fast convergence and slow convergence, the topological classification of sets of subsums of real series.
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Week 3 (March 4 and 6): The Guthrie-Nymann Cantorval and the center of distances: the Guthrie-Nymann Cantorval, topological properties of the center of distances, center of distances for achievement sets, non-achievable Cantorvals.
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Week 4 (March 11 and 13): Multigeometric series and sets of weighted sums: multigeometric series and sufficient conditions for Cantorvals, geometric weighted sums, topological characterization of sets of weighted sums, Kakeya conditions.
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Week 5 (March 18 and 20) Algebraic operations on achievement sets and recovering sequences: algebraic sum and difference of central Cantor sets, algebraic sums of general achievement sets, recovering the sequence from its achievement set.
