Characterization and solvability of the Fragment System

Characterize the set S_F of radii r in [0, chg(F)) for which the Fragment System associated with a finite subset F of ℓ₁^N has a unique solution, or establish a sufficient condition for membership in S_F; determine whether the Fragment System is solvable for every r in [0, chg(F)), equivalently whether S_F = [0, chg(F)).

Background

The paper introduces the Fragment System, a finite system of linear equations whose coefficients determine the weight measure of a union of sufficiently small coordinate-aligned cubes centered at a finite subset F of ℓ₁^N. The set S_F consists of those radii r in [0, chg(F)) for which the coefficient matrix of this system is invertible.

The authors prove that the Fragment System has a unique solution for all sufficiently small radii near 0. This local solvability is enough to establish continuity of magnitude at every finite subset of ℓ₁^N. However, the weight-measure formula is stated for all radii in S_F, and the paper does not determine whether solvability persists throughout the entire interval [0, chg(F)).