Another important example is the function formula_32, where formula_33 is a real number. For this function, the relevant series diverges and so Khinchin's theorem tells us that almost every number is formula_34-approximable. This is the same as saying that every such number is "well approximable", where a number is called well approximable if it is not badly approximable. So an appropriate analogue of the Jarník-Besicovitch theorem should concern the Hausdorff dimension of the set of badly approximable numbers. And indeed, V. Jarník proved that the Hausdorff dimension of this set is equal to one. This result was improved by W. M. Schmidt, who showed that the set of badly approximable numbers is "incompressible", meaning that if formula_35 is a sequence of bi-Lipschitz maps, then the set of numbers "x" for which formula_36 are all badly approximable has Hausdorff dimension one. Schmidt also generalized Jarník's theorem to higher dimensions, a significant achievement because Jarník's argument is essentially one-dimensional, depending on the apparatus of continued fractions. Another topic that has seen a thorough development is the theory of uniform distribution mod 1. Take a sequence "a"1, "a"2, ... of real numbers and consider their "fractional parts". That is, more abstractly, look at the sequence in R/Z, which is a circle. For any interval "I" on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer "N", and compare it to the proportion of the circumference occupied by "I". "Uniform distribution" means that in the limit, as "N" grows, the proportion of hits on the interval tends to the 'expected' value. Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout analytic number theory in the bounding of error terms. Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature. There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the "Littlewood conjecture" and the "Lonely runner conjecture". It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.