The Local-Global Principle for Integral Soddy Sphere Packings
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Fix an integral Soddy sphere packing P. Let K be the set of all curvatures in P. A number n is called represented if n is in K, that is, if there is a sphere in P with curvature equal to n. A number n is called admissible if it is everywhere locally represented, meaning that n is in K(mod q) for all q. It is shown that every sufficiently large admissible number is represented.
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Cited by 2 Pith papers
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Poissonian correlations of $\alpha n^d$ mod $1$
α n^d mod 1 exhibits Poissonian ℓ-point correlations for almost all α when d is large (depending on ℓ) and for a full-dimensional set of badly approximable α via determinant method counting and Fourier transference.
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Poissonian correlations of $\alpha n^d$ mod $1$
For large d, the sequence α n^d mod 1 has Poissonian ℓ-point correlations for almost all α and for a full-dimensional set of badly approximable α via determinant-method counting on hypersurfaces.
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