I thank Dan Hoey for pointing out a basic mistake in my statement << Define the n-onion O_n as the union of all spheres S_k(n) about 0 in R^n whose radius is sqrt(k) for any integer k >= 0 (including the degenerate sphere S_0(n)).

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The last statement should read: Then because of the sharpness of the 4-square theorem, for each n = 1,2,3, some spheres S_k(n) will not intersect Z^n, while for n >= 4,every S_k(n) will intersect Z^n. Dan A.