can be further enlarged. The values which may be generated in this way are Lagrange numbers, which are part of the Lagrange spectrum. They converge to the number 3 and are related to the Markov numbers.See Michel Waldschmidt: Introduction to Diophantine methods irrationality and transcendence , pp 24–26. Khinchin's theorem on metric Diophantine approximation and extensions Let be a positive real-valued function on positive integers (i.e., a positive sequence) such that is non-increasing. A real number x (not necessarily algebraic) is called -approximable if there exist infinitely many rational numbers p/q such that Aleksandr Khinchin proved in 1926 that if the series diverges, then almost every real number (in the sense of Lebesgue measure) is -approximable, and if the series converges, then almost every real number is not -approximable. The circle of ideas surrounding this theorem and its relatives is known as metric Diophantine approximation or the metric theory of Diophantine approximation (not to be confused with height "metrics" in Diophantine geometry) or metric number theory. proved a generalization of Khinchin's result, and posed what is now known as the Duffin–Schaeffer conjecture on the analogue of Khinchin's dichotomy for general, not necessarily decreasing, sequences . proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture. Hausdorff dimension of exceptional sets An important example of a function to which Khinchin's theorem can be applied is the function , where c > 1 is a real number. For this function, the relevant series converges and so Khinchin's theorem tells us that almost every point is not -approximable. Thus, the set of numbers which are -approximable forms a subset of the real line of Lebesgue measure zero. The Jarník-Besicovitch theorem, due to V. Jarník and A. S. Besicovitch, states that the Hausdorff dimension of this set is equal to . In particular, the set of numbers which are -approximable for some (known as the set of very well approximable numbers) has Hausdorff dimension one, while the set of numbers which are -approximable for all (known as the set of Liouville numbers) has Hausdorff dimension zero. Another important example is the function , where is a real number. For this function, the relevant series diverges and so Khinchin's theorem tells us that almost every number is -approximable. This