Ramsey Properties of Subsets of N Solvability of Linear Diophantine Systems within Wm Sets 16
نویسنده
چکیده
With my appreciation, I wish to thank my advisor Prof. Hillel Furstenberg for his guidance, support and encouragement during these unforgettable years of learning and research Abstract We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain " algebraic patterns " in every subset in the family. By " algebraic pattern " we mean a set of solutions of a system of diophantine equations. In this work we investigate a concrete family of subsets-WM sets. These sets are characterized by the property that the dynamical systems associated to such sets are " weakly mixing " , and as such they represent a broad family of randomly constructed subsets of N. We find that certain systems of equations are solvable within every WM set, and our subject is to learn which systems have this property. We give a complete characterization of linear diophantine systems which are solvable within every WM set. In addition we study some non-linear equations and systems of equations with regard to the question of solvability within every WM set.
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Solvability of linear equations within the weak mixing sets
We introduce a new class of “random” subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We provide necessary and sufficient conditions of solvability of systems of linear equations within every WM subset of N. We also provide a proof of solvability of partition-regular systems (Rado’s systems) within every WM su...
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