On kk-point configuration sets with nonempty interior

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in $\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb R^2$, and dot products of differences in $\mathbb R^d$.Comment: 32 pages, no figure

The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{\alpha,A}((z,t)) = \left(|z|^\alpha + A |t|^{\alpha/2}\right)^{1/\alpha}$, for $\alpha \ge 2$ and $A>0$. This natural family includes the canonical Cygan-Kor\'anyi norm, corresponding to $\alpha =4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $\alpha=2$ is the best possible, in all dimensions

Constant gap length trees in products of thick Cantor sets

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of admissible gap lengths has non-empty interior. This builds on the authors' previous work on distance sets of products of Cantor sets of sufficient Newhouse thickness

Fourier integral operators, fractal sets, and the regular value theorem

AbstractWe prove that if E⊂R2d, for d⩾2, is an Ahlfors–David regular product set of sufficiently large Hausdorff dimension, denoted by dimH(E), and ϕ is a sufficiently regular function, then the upper Minkowski dimension of the set{w∈E:ϕl(w)=tl;1⩽l⩽m} does not exceed dimH(E)−m, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry