Davenport Constants of High-Dimensional Integer Sets

The Davenport constant of a finite set of integer vectors is the maximum length of an irreducible zero-sum sequence over that set. For nonempty sets closed under negation, it equals the largest ₁-norm of a Graver basis element of the matrix formed by those vectors. We study extremal and average-case versions in growing dimension, with applications to integer programming. For integer cubes [-R,R]ᵈ ∩ Zᵈ and the sign cube \-1,1\ᵈ, totally unimodular encodings improve on the upper bounds from direct circuit decomposition, while random perturbations of Hadamard matrices give new lower bounds. A key ingredient is a uniform affine Bernoulli variant of Nguyen and Wood's surjectivity theorem. In every sufficiently large dimension the sign-cube lower bound is dd/22⁻ᵈ⁻ᵒ⁽ᵈ⁾, and the integer-cube lower bound is Rᵈ dd/22⁻ᵒ⁽ᵈ⁾ uniformly for polynomially bounded integer radii. For random sign-cube sequences of even length conditioned on summing to zero, we prove a sharp threshold from irreducibility to reducibility at dlog d/log 4, far below the extremal scale. For fixed p and R, we determine exact exponential rates for lattice ₚ-balls when $p=1$, R≥3, or 2≤ p<∞, Rᵖ≥3. For each fixed integer k≥3, we determine the exponential rate of the worst-case ₁-distance from an optimal LP vertex to the nearest integer optimum in standard-form programs with ternary matrices having at most k nonzero entries per column.