Recursion Theory and Diophantine Approximation

点击翻译按钮获取中文摘要

Part of our understanding mathematical objects and their behaviors involves knowing the ways by which we specify those objects and the means that we employ to study them. For example, the question of whether there are infinitely many 7's in the decimal expansion of pi asks about a property of the geometric constant pi as expressed in our base-10 representation of it. This research project in the foundations of mathematics investigates the effective, and more generally definable, aspects of Diophantine approximation, the meta-mathematical status of familiar theorems in countable combinatorics, and the pure structure theory of relative definability. One can view classical theorems in Diophantine approximation, such as Borel's almost-every real number is absolutely normal or Roth's every irrational algebraic number has irrationality exponent 2, as asserting properties of real numbers in terms of our descriptions of them. Borel's theorem asserts a property of expansions by integer bases, and Roth's theorem asserts a property of approximation by rational numbers. There is a natural affinity between this tradition in number theory and the study of definability in recursion theory. This research project will investigate the connections between these areas. For example, what is the exact relationship between the irrationality exponent of a real number, or more generally its Mahler transcendence measures, and the Kolmogorov complexity of the initial segments of its binary expansion? Second, a real number is absolutely normal if for every integer base b, each digit appears with asymptotic frequency 1/b. What other patterns of asymptotic frequencies are possible? This last question also touches dynamical systems in the form of Furstenberg's x2 x3 conjecture.