10 papers
Prescribed projections for a general class of maps and efficient covering by variable plane curves
Alan Chang, Robert Fraser, Alex McDonald +2
We prove a prescribed projection theorem for any class of non-linear projections defined on open subsets of the plane which satisfies some mild geometric conditions. This generaliz…
Arithmetic-progression gap sets in Cantor sets
Samantha Sandberg-Clark, Krystal Taylor, Alexia Yavicoli
We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set , we investigate not only w…
Applications of Nonlinear Projections to Rectifiable 1-sets
Rosemarie Bongers, Paige Bright, Caleb Marshall +1
Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $…
Favard length and generalized projections
Izabella Laba, Alex McDonald, Krystal Taylor
We investigate generalized Favard lengths associated to smooth families of nonlinear projections. Under suitable regularity and transversality assumptions, we prove that generalize…
A Quantified Two-projection Theorem for Nonlinear Projections
Zhangze Li, Krystal Taylor
The classic Besicovitch projection theorem asserts that if a set is purely -unrectifiable with finite length in , its orthogonal projection has Lebesgue measure ze…
Interior of distance trees over thin Cantor sets
Yeonwook Jung, Krystal Taylor
It is known that if a compact set in has Hausdorff dimension greater than , then its -chain distance set $$Δ^n(E) = \left\{\left(\left|x^1-x^2\right|…
Triangles in the Plane and arithmetic progressions in thick compact subsets of
Samantha Sandberg-Clark, Krystal Taylor
This article focuses on the occurrence of 3-point configurations in subsets of of sufficient thickness. We prove that a compact set contains…
Counting Lattice Points near Korányi Spheres via Generalized Radon Transforms
Rajula Srivastava, Krystal Taylor
In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law.…
Point configurations in sets of sufficient topological structure and a topological {E}rdős similarity conjecture
Alex McDonald, Krystal Taylor
We explore the occurrence of point configurations within non-meager (second category) Baire sets. A celebrated result of Steinhaus asserts that and contain an interval…
Prescribed projections and efficient coverings by curves in the plane
Alan Chang, Alex McDonald, Krystal Taylor
Davies efficient covering theorem states that an arbitrary measurable set in the plane can be covered by full lines so that the measure of the union of the lines has the same m…