Effective Results in The Metric Theory of Quantitative Diophantine Approximation

Abstract

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results does not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. This paper addresses this problem and makes the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most $δ>0$, an arbitrarily small constant chosen in advance. We deduce from this the fully effective results for Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative results on $M_{0}$-sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers.