Ruben Carpenter

It is a classical fact that every \(n\)-element set of positive reals has at least \(\binom{n+1}{2}+1\) distinct subset sums, with equality exactly for homogeneous arithmetic progressions when \(n \geq 4\). We establish stability versions of this inverse theorem in two regimes. First, for any parameter \(M \leq n-4\), we precisely characterize the \(n\)-element sets of positive reals with at most \(\binom{n+1}{2}+1+M\) subset sums. Second, for any constant \(C\), we provide a characterization, sharp up to constants, of the \(n\)-element sets of positive reals with at most \(Cn^2\) distinct subset sums. Along the way, we constrain, for any fixed \(d \geq 2\), the structure of \(n\)-element subsets of \(\mathbb{R}^d\) with \(o(n^{d+1})\) subset sums.

For distinct real numbers \(a_1,\ldots,a_n\) and distinct real numbers \(b_1,\ldots,b_n\), consider the sum \(S=\sum_{i=1}^n a_i b_{\pi(i)}\) as \(\pi\) ranges over the permutations of \([n]\). We show that this sum always assumes at least \(\Omega(n^3)\) distinct values, which is optimal. This support bound complements recent work of Do, Nguyen, Phan, Tran, and Vu on the anticoncentration properties of \(S\) when \(\pi\) is chosen uniformly at random.

The random billiard walk is a stochastic process \((L_t)_{t\geq 0}\) in which a laser moves through the Coxeter arrangement of an affine Weyl group in \(\mathbb{R}^d\), reflecting at each hyperplane with probability \(p\in(0,1)\) and transmitting unchanged otherwise. Defant, Jiradilok, and Mossel introduced this process from the perspective of algebraic combinatorics and established that, for initial directions aligned with the coroot lattice, \(L_t/\sqrt{t}\) converges to a centered spherical Gaussian. We bring analytic tools from ergodic theory and probability to the problem and extend this central limit theorem to all initial directions. More strongly, we prove the rescaled trajectories \(t\mapsto n^{-1/2}L_{tn}\) converge to isotropic Brownian motion. Away from directions with rational dependencies, the limiting covariance varies continuously in \(p\) and the initial direction.