View-obstruction problems. III. (English) Zbl 0563.10026

Cusick, T. W.; Pomerance, Carl

J. Number Theory 19, 131-139 (1984).

[Part II, cf. Proc. Am. Math. Soc. 84, 25-28 (1982; Zbl 0474.10023).]
Suppose C is a closed convex body in \(E^ n\) which contains the origin as an interior point. Define \(\alpha\) C for each real number \(\alpha\geq 0\) to be the magnification of C by the factor \(\alpha\) 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 \(\Delta\) (C,\(\alpha)\) by \(\Delta (C,\alpha)=\{\alpha C+(m_ 1+,...,m_ n+):\) \(m_ 1,...,m_ n\) 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 \(\alpha\) such that any half-line L given by \(x_ i=a_ it\) \((i=1,2,...,n)\), where the \(a_ i\) (1\(\leq i\leq n)\) are positive real numbers and the parameter t runs through [0,\(\infty)\), intersects \(\Delta\) (C,\(\alpha)\).
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. 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 \(\| w_ ix\| \geq (n+1)^{-1}\).

MSC:

11H06	Lattices and convex bodies (number-theoretic aspects)
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A30	Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)

Keywords:

magnification; translation; n-dimensional cube; dimension 4; system of simultaneous congruences

Citations:

Zbl 0265.52003; Zbl 0273.10025; Zbl 0474.10023

References:

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