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Abstract. The counting and (upper) mass dimensions of a set A ⊆ R d arewhere ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ R d with side length C → ∞. We give a characterization of the counting dimension via coverings:in which the infimum is taken over cubic coverings {C i } of A ∩ C. Then we prove Marstrand-type theorems for both dimensions.

Denote by PS(α) the image of the Piatetski-Shapiro sequence n → ⌊n α ⌋, where α > 1 is non-integral and ⌊x⌋ is the integer part of x ∈ R. We partially answer the question of which bivariate linear equations have infinitely many solutions in PS(α): if a, b ∈ R are such that the equation y = ax + b has infinitely many solutions in the positive integers, then for Lebesgue-a.e. α > 1, it has infinitely many or at most finitely many solutions in PS(α) according as α < 2 (and 0 ≤ b < a) or α > 2 (and (a, b) = (1, 0)). We collect a number of interesting open questions related to further results along these lines.

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual set of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along N and along cosets of multiplicative subsemigroups of N, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.

By juxtaposing ideas from fractal geometry and dynamical systems, Hillel Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce analogues of some of the notions and results surrounding Furstenberg's work in the discrete setting of the integers. In particular, we define a new class of fractal sets of integers that parallels the notion of ×r-invariant sets on the 1-torus, and we investigate the additive independence between these fractal sets when they are structured with respect to different bases. We obtain• an integer analogue of a result of Furstenberg regarding the classification of all sets that are simultaneously ×2 and ×3 invariant (see Theorem B); • an integer analogue of a result of Lindenstrauss-Meiri-Peres on iterated sumsets of ×r-invariant sets (see Theorem C); • an integer analogue of Hochman and Shmerkin's solution to Furstenberg's sumset conjecture regarding the dimension of the sumset X + Y of a ×r-invariant set X and a ×s-invariant set Y (see Theorem D). To obtain the latter, we provide a quantitative strengthening of a theorem of Hochman and Shmerkin which provides a lower bound on the dimension of λX + ηY uniformly in the scaling-parameters λ and η at every finite scale (see Theorem A). Our methods yield a new combinatorial proof of the theorem of Hochman and Shmerkin that avoids the machinery of local entropy averages and CP-processes.

We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given.

Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers (ψn) n∈N and differentiable functions (ϕn : J → R) n∈N so that for Lebesgue-a.e. θ ∈ J, the inequality nθ + ϕn(θ) ≤ ψn has infinitely many solutions. The main novelty is that the magnitude of the perturbation |ϕn(θ)| is allowed to exceed ψn, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main result, we prove that if the linear equation y = ax + b, a, b ∈ R, has infinitely many solutions in N, then for Lebesgue-a.e. α > 1, it has infinitely many or finitely many solutions of the form ⌊n α ⌋ according as α < 2 or α > 2.

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r/ log s is irrational and X and Y are ×rand ×sinvariant subsets of [0, 1], respectively, then dimOur main result yields information on the size of the sumset λX + ηY uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.