Abstract. We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each s there is a class of sets such that a measure has modified Fourier dimension greater than or equal to s if and only if it annihilates all sets in the class.

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We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in R d whose centres are independent, identically distributed random variables. The formulas obtained involve the rate of decrease of the radii of the balls and multifractal properties of the measure according to which the balls are distributed, and generalise formulas that are known to hold for particular classes of measures.

The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$. This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$-equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha)$. In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^m$-diffeomorphisms for any $m$.Comment: Minor improvements of expositio