LAUSR

I recently discovered that the proof of Proposition 3.1 in my 2012 paper 1[1] (stated on page 522) is not correct. The proof can be corrected but this introduces a factor on the left-hand side of Equation (3.2). This factor is not important for Proposition 3.1 itself but it is important for its applications. Unfortunately, it invalidates the end of the proof of Theorem 1.1. I could not find a complete correction for it, and I present a slightly weaker statement below. On the other hand, the statements of Theorems 1.2 and 1.3 are not modified because their proofs are not affected in a crucial way by this factor. I use the notation and labels of the paper. These labels are of the form (a.b) with a ≥ 1. There will be no confusion with the labels of this corrigendum which are the form (a.b) with a = 0. For the properties of continued fractions, I refer to §1.1 of the paper. Correction to the proof of Proposition 3.1 In Equation (3.2) of Proposition 3.1, the implicit constant in the symbol does not depend on m but it depends on α, f, s and more importantly on k. The dependence on k was not noticed in the paper because of the error in the proof of Equation (3.2). Moreover, the independence of Equation (3.2) on k was explicitly used many times in the proofs of the main results of the paper, so that many arguments now have to be corrected. We are given f(x) defined on R \ (πZ) and such that |f(x)| ≤ c/| sin(x)|r. The statement of Proposition 3.1 remains the same except that, for applications, Equation (3.2) must be made more precise. This is done as follows: For every α ∈ (0, 1) \ Q, s > r ≥ 1, m ≥ 0, k ≥ 0 and N ≥ 1 such that qm ≤ N < qm+1, we have