We prove the following conjecture of Furstenberg (1969): if 𝐴,𝐵⊂[0,1] are closed and invariant under ×𝑝 mod1 and ×𝑞 mod1, respectively, and if log𝑝/log𝑞∉ℚ, then for all real numbers 𝑢 and 𝑣, dim_H (𝑢𝐴 + 𝑣) ∩ 𝐵 ≤ max {0, dim_H 𝐴 + dim_H 𝐵 − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on ℝ. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.

Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined L^p spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

We study a family of maximal operators that provides a continuous link connecting the Hardy–Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.

Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.