Quantum Discrepancy: A Non-Commutative Version of Combinatorial Discrepancy

In this paper, we introduce a notion of quantum discrepancy, a non-commutative version of combinatorial discrepancy which is defined for projection systems, i.e. finite sets of orthogonal projections, as non-commutative counterparts of set systems. We show that besides its natural algebraic formulation, quantum discrepancy, when restricted to set systems, has a probabilistic interpretation in terms of determinantal processes. Determinantal processes are a family of point processes with a rich algebraic structure. A common feature of this family is the local repulsive behavior of points. Alishahi and Zamani (2015) exploit this repelling property to construct low-discrepancy point configurations on the sphere. We give an upper bound for quantum discrepancy in terms of $N$, the dimension of the space, and $M$, the size of the projection system, which is tight in a wide range of parameters $N$ and $M$. Then we investigate the relation of these two kinds of discrepancies, i.e. combinatorial and quantum, when restricted to set systems, and bound them in terms of each other.

Discrepancy of Cartesian Products of Arithmetic Progressions

We determine the combinatorial discrepancy of the hypergraph ${\cal H}$ of cartesian products of $d$ arithmetic progressions in the $[N]^d$–lattice ($[N] = \{0,1,\ldots,N-1\}$). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden's theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for $d$–dimensional arithmetic progressions by proving ${\rm disc}({\cal H}) = \Theta(N^{d/4})$ for every fixed integer $d \ge 1$. This extends the famous lower bound of $\Omega(N^{1/4})$ of Roth (1964) and the matching upper bound $O(N^{1/4})$ of Matoušek and Spencer (1996) from $d=1$ to arbitrary, fixed $d$. To establish the lower bound we use harmonic analysis on locally compact abelian groups. For the upper bound a product coloring arising from the theorem of Matoušek and Spencer is sufficient. We also regard some special cases, e.g., symmetric arithmetic progressions and infinite arithmetic progressions.