The uniformity invariant for Lebesgue measure is defined to be the least cardinal of a non-measurable set of reals, or, equivalently, the least cardinal of a set of reals which is not Lebesgue null. This has been studied intensively for the past 30 years and much of what is known can be found in [?] and other standard sources. Among the well known results about this cardinal invariant of the co...

We study the almost Mathieu operator: 1) + cos(2n +)u(n), on l 2 (Z), and show that for all ; , and (Lebesgue) a.e. , the Lebesgue measure of its spectrum is precisely j4 ? 2jjj. In particular, for jj = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational 's (and jj = 2) we show that the Hausdorr dimension of the spectrum is smaller than or equal to 1=2.

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for Gδ ...

The ordinary notion of algorithmic randomness of reals can be characterised as MartinLöf randomness with respect to the Lebesgue measure or as Kolmogorov randomness with respect to the binary representation. In this paper we study the question how the notion of algorithmic randomness induced by the signed-digit representation of the real numbers is related to the ordinary notion of algorithmic ...

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and stron...

Polynomial interpolation is a classical method to approximate continuous functions by polynomials. To measure the correctness of the approximation, Lebesgue constants are introduced. For a given node system X = {x1 < . . . < xn+1} (xj ∈ [a, b]), the Lebesgue function λn(x) is the sum of the modulus of the Lagrange basis polynomials built on X. The Lebesgue constant Λn assigned to the function λ...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

We consider the classical Vitali's construction of nonmeasurable subsets of the real line R and investigate its analogs for various uncountable subgroups of R. Among other results we show that if G is an uncountable proper analytic subgroup of R then there are Lebesgue measurable and Lebesgue nonmeasurable selectors for R/G . 0. Introduction In this paper we investigate some properties of selec...

Abstract. We consider a smooth two-parameter family fa,L : θ 7→ θ+a+LΦ(θ) of circle maps with a finite number of critical points. For sufficiently large L we construct a set A (∞) L of a-values of positive Lebesgue measure for which the corresponding fa,L exhibits an exponential growth of derivatives along the orbits of the critical points. Our construction considerably improves the previous on...

The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publicatio...