This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb {R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb {R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided that A satisfies a mild Yavicoli-thickness condition and an r-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided that the Newhouse thickness of A is at least $1$. Moreover, we prove that compact sets $A\subset \mathbb {R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided A satisfies a mild Yavicoli-thickness condition and an r-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of C is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.

Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.