Abstract: The classical Borsuk–Ulam theorem states that any continuous map from the d-sphere to d-space identifies two antipodal points. Over the last 90 years numerous applications of this result across mathematics have been found. Yang and later Gromov proved versions of the Borsuk–Ulam theorem for lower dimensional codomains that quantify the size of the largest fiber. Dually, I will present a Borsuk–Ulam theorem for higher dimensional codomains that gives approximate coincidences. I will present some applications of these “overdetermined” Borsuk–Ulam theorems such as results about the structure of zeros of trigonometric polynomials, a proof of a 1971 conjecture that any closed spatial curve inscribes a parallelogram, and finding well-behaved smooth functions to the unit circle in any closed finite codimension subspace of square-intergrable complex functions.