Harmonic functions on Z^d with u(0) nonzero have at least (10^(-10)/d) n^2 nonzero values in each n-cube, and this n^2 order is sharp in dimension three.
The geometry of Hilbert functions
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abstract
We compare and contrast results of E. Davis, of A. Bigatti, A.V. Geramita and the author, and of J. Ahn and the author. The underlying idea is that certain numerical conditions on the Hilbert function of a finite set of points in projective space force geometric consequences on the base loci of the linear systems of hypersurfaces containing the points. When the points have uniform position, this tends to force better behavior for these base loci. However, unexpected behavior is still possible, and we give examples.
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Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity
Harmonic functions on Z^d with u(0) nonzero have at least (10^(-10)/d) n^2 nonzero values in each n-cube, and this n^2 order is sharp in dimension three.