Arithmetic properties of Heilbronn sums

AbstractFor each odd prime q an integer NHq (NH3 = −1, NH5 = −1, NH7 = 97, NH11 = −243, …) is defined as the norm from L to Q of the Heilbronn sum Hq = TrIQ(ζ)(ζ), where ζ is a primitive q2th root of unity and L ⊃- Q(ζ) the subfield of degree q. Various properties are proved relating the congruence properties of Hq and NHq modulo p (p ≠ q prime) to the Fermat quotient (pq − 1 − 1)q (mod q); in particular, it is shown that NHq is even iff 2q − 1 ≡ 1 (mod q2)

Symmetries and Ramsey properties of trees

AbstractIn this paper we show the extent to which a finite tree of fixed height is a Ramsey object in the class of trees of the same height can be measured by its symmetry group

How constructive is constructing measures?

Given some set, how hard is it to construct a measure supported by it? We classify some variations of this task in the Weihrauch lattice. Particular attention is paid to Frostman measures on sets with positive Hausdorff dimension. As a side result, the Weihrauch degree of Hausdorff dimension itself is determined.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Applications of Effective Probability Theory to Martin-Löf Randomness

The original publication is available at www.springerlink.comInternational audienceWe pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications. (i) We prove a general version of Birkhoff's ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable. This result significantly improves V'yugin and Nandakumar's ones. (ii) We provide a general framework for deriving sharper theorems for random points, sensitive to the speed of convergence. This offers a systematic approach to obtain results in the spirit of Davie's ones. (iii) Proving an effective version of Prokhorov theorem, we positively answer a question recently raised by Fouché: can random Brownian paths reach any random number? All this shows that layerwise computability is a powerful framework to study Martin-Löf randomness, with a wide range of applications