View-obstruction problems, III

AbstractSuppose C is a closed convex body in En which contains the origin as an interior point. Define αC for each real number α ≥ 0 to be the magnification of C by the factor α and define C + (m1, …, mn) for each point (m1, …, mn) in En to be the translation of C by the vector (m1, …, mn). Define the point set Δ(C, α) by Δ(C, α) = {αC + (m1 + 12, …, mn + 12): m1, …, mn nonnegative integers}. The view-obstruction problem for C is the problem of finding the constant K(C) defined to be the lower bound of those α such that any half-line L given by xi = ait (i = 1, 2, …, n), where the ai (1 ≤ i ≤ n) are positive real numbers and the parameter t runs through [0, ∞), intersects Δ(C, α). The paper considers the case where C is the n-dimensional cube with side 1, and in this case the constant K(C) is evaluated for n = 4. The proof in dimension 4 depends on a theorem (proved via exponential sums) concerning the existence of solutions for a certain system of simultaneous congruences. The proofs in dimensions 2 and 3 are much simpler, and for these dimensions several other proofs have previously been given. For real x, let |x| denote the distance from x to the nearest integer. A non-geometric description of our principal result is that we prove the case n = 4 of the following conjecture: For any n positive integers w1, …, wn there is a real number x such that each |wix| ≥ (n + 1)−1