Let h be a harmonic function on ℝn of suitably restricted growth. It is known
that if h vanishes, or is bounded, on the lattice ℤn−1 × {0},
then the same is true on ℝn−1 × {0}.
This paper presents sharp results which show that, if n [ges ] 3, then the
same conclusions can be drawn even if information about h is missing on a substantial
proportion of the lattice points. As corollaries we obtain uniqueness and Liouville-type
theorems for harmonic, and also polyharmonic, functions which improve results
by several authors.