View-obstruction problems. II

Author:
T. W. Cusick

Journal:
Proc. Amer. Math. Soc. **84** (1982), 25-28

MSC:
Primary 10F10; Secondary 52A43, 52A45

DOI:
https://doi.org/10.1090/S0002-9939-1982-0633270-4

MathSciNet review:
633270

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Abstract: Let denote the region of -dimensional Euclidean space . Suppose is a closed convex body in which contains the origin as an interior point. Define for each real number to be the magnification of by the factor and define for each point in to be the translation of by the vector . Define the point set by nonnegative integers}. The *view-obstruction problem for* is the problem of finding the constant defined to be the lower bound of those such that any half-line given by , where the are positive real numbers, and the parameter runs through , intersects .

The paper considers the case where is the -dimensional cube with side 1, and in this case the constant is known for . The paper gives a new proof for the case . Unlike earlier proofs, this one could be extended to study the cases with .

**[1]**U. Betke and J. M. Wills,*Untere Schranken für zwei diophantische Approximations-Funktionen*, Monatsh. Math.**76**(1972), 214–217 (German). MR**0313194**, https://doi.org/10.1007/BF01322924**[2]**T. W. Cusick,*View-obstruction problems*, Aequationes Math.**9**(1973), 165–170. MR**0327665**, https://doi.org/10.1007/BF01832623**[3]**T. W. Cusick,*View-obstruction problems in 𝑛-dimensional geometry*, J. Combinatorial Theory Ser. A**16**(1974), 1–11. MR**0332539****[4]**I. J. Schoenberg,*Extremum problems for the motions of a billiard ball. II. The 𝐿_{∞} norm*, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math.**38**(1976), no. 3, 263–279. MR**0405247**

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0633270-4

Article copyright:
© Copyright 1982
American Mathematical Society