We study different conditions on the set of roots of the Fourier transform of a measure
on the Euclidean space, which yield that the measure is absolutely continuous with
respect to the Lebesgue measure. We construct a monotone sequence in the real line with
this property. We construct a closed subset of ${\mathbb{R}}^{d}$ which contains a lot of
lines of some fixed direction, with the property that every measure with spectrum
contained in this set is absolutely continuous. We also give examples of sets with such
property...

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb{T}$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb{T}$, and Bohr if it is recurrent for all products of rotations on $\mathbb{T}$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which...

A measure is called ${L}^{p}$-improving if it acts by convolution as a bounded operator from ${L}^{p}$ to ${L}^{q}$ for some q > p. Positive measures which are ${L}^{p}$-improving are known to have positive Hausdorff dimension. We extend this result to complex ${L}^{p}$-improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of ${L}^{p}$-functions.

We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...

We describe necessary and sufficient conditions for a fixed point free planar homeomorphism that preserves the standard Reeb foliation to embed in a planar flow that leaves the foliation invariant.

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