This paper deals with better-than-uniform approximation by superharmonic functions on non-compact sets. An application of the main result improves recent work of GowriSankaran and Nersessian concerning approximation with an error bound that decays arbitrarily quickly at infinity.

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.

If u is a superharmonic function on ℝ2, then

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for all (x, y) ∈ ℝ2. This follows from the fact that a line segment in ℝ2 is non-thin at each of its constituent points. (See Doob [1, 1.XI] or Helms [7, Chapter 10] for an account of thin sets and the fine topology.) The situation is different in higher dimensions. For example, if u is the Newtonian potential on ℝ3 defined by

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then

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Corollary 2 below will show that, nevertheless, for nearly every vertical line L, the value of a superharmonic function at any point X of L is determined by its lower limit along L at X.

Throughout this paper, we let n [ges ] 3. A typical point of ℝn will be denoted by X or (X′, x), where X′ ∈ ℝn−1 and x ∈ ℝ. Given any function f[ratio ]ℝn → [−∞, +∞] and any point X, we define the vertical cluster set of f at X by

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and the fine cluster set of f at X by

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The fine topology on ℝn (n[ges ]2) is the coarsest topology for which all superharmonic functions on ℝn are continuous. We refer to Doob [11, 1.XI] for its basic properties and its relationship to the notion of thinness. This paper presents several theorems relating the fine topology to limits of functions along parallel lines. (Results of this nature for the minimal fine topology have been given by Doob – see [10, Theorem 3.1] or [11, 1.XII.23] – and the second author [15].) In particular, we will establish improvements and generalizations of results of Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihl and Seidel [6], Schneider [21], Berman [7], and Armitage and Nelson [4], and will also solve a problem posed by the latter authors.

An early version of our first result is due to Evans [12, p. 234], who proved that, if u is a superharmonic function on ℝ3, then there is a set E⊆ℝ2×{0}, of two-dimensional measure 0, such that u(x, y,·) is continuous on ℝ whenever (x, y, 0)∉E. We denote a typical point of ℝn by X=(X′ x), where X′∈ℝn−1 and x∈ℝ. Let π:ℝn→ℝn−1×{0} denote the projection map given by π(X′, x) = (X′, 0). For any function f:ℝn→[−∞, +∞] and point X we define the vertical and fine cluster sets of f at X respectively by

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and

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Sets which are open in the fine topology will be called finely open, and functions which are continuous with respect to the fine topology will be called finely continuous. Corollary 1(ii) below is an improvement of Evans' result.

This paper shows that a subharmonic function in the half-space which does not grow too rapidly near the boundary and which does not have asymptotic value +∞ at too many points must have finite minimal fine limits at a boundary set of positive measure. For harmonic functions, the conclusion may be expressed in terms of non-tangential limits. A related Phragmén–Lindelöf theorem is also established.

The purpose of this paper is to answer some questions posed by Doob [2] in 1965 concerning the boundary cluster sets of harmonic and superharmonic functions on the half-space D given by D = ℝn−1 × (0, +∞), where n[ges ]2. Let f[ratio ]D→[−∞, + ∞] and let Z∈δD. Following Doob, we write BZ (respectively CZ) for the non-tangential (respectively minimal fine) cluster set of f at Z. Thus l∈BZ if and only if there is a sequence (Xm) of points in D which approaches Z non-tangentially and satisfies f(Xm)→l. Also, l∈CZ if and only if there is a subset E of D which is not minimally thin at Z with respect to D, and which satisfies f(X)→l as X→Z along E. (We refer to the book by Doob [3, 1.XII] for an account of the minimal fine topology. In particular, the latter equivalence may be found in [3, 1.XII.16].) If f is superharmonic on D, then (see [2, §6]) both sets BZ and CZ are subintervals of [−∞, + ∞]. Let λ denote (n−1)-dimensional measure on δD. The following results are due to Doob [2, Theorem 6.1 and p. 123].