View-obstruction problems, III

Abstract

Suppose C is a closed convex body in E n 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 + ( m 1, …, m n ) for each point ( m 1, …, m n ) in E n to be the translation of C by the vector ( m 1, …, m n ). Define the point set Δ( C, α) by Δ(C, α) = {αC + (m 1 + 1 2 , …, m n + 1 2 ): m 1, …, m n 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 x i = a i t ( i = 1, 2, …, n), where the a i (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 w 1, …, w n there is a real number x such that each | w i x| ≥ ( n + 1) −1.