Let B be an almost-periodic
(a.p.) function with mean value zero. Let G(t) =∫0tB(s)ds. The well-known
theorem of Bohr states that G(t) is uniformly bounded iff G(t) is a.p. This theorem
may be reformulated in the following way. Let Ω be the hull of B, and let (Ω,R) be
the flow on Ω defined by translation. Since B is a.p., Ω is a compact abelian
topological group. There is a continuous b : Ω → R and an ω0∈ Ω such that
b(ω0⋅ t) = B(t). I.e., b “extends B to Ω”. Then Bohr’s theorem is equivalent to the
following: G(t) is bounded iff there is a continuous r : Ω → R such that
r(ω ⋅ t) − r(ω) =∫0tb(ω ⋅ s)ds(ω ∈ Ω,t ∈ R).
In this paper, we consider the case when G(t) is unbounded. Two results
are obtained. The first is a generalization of Bohr’s theorem: let μ be
(normalized) Haar measure on Ω, and let gω(t) =∫0tb(ω ⋅ s)ds(ω ∈ Ω,t ∈ R);
then n→∞1∕2nγ{t ∈ [−n,n]|gω(t) ∈ I} > 0 for some compact I ⊂ R
and some ω ∈ Ω iff there exists a μ-measurable r : Ω → R such that
r(ω ⋅ t) − r(ω) =∫0tb(ω ⋅ s)ds(ω ∈ Ω,t ∈ R). Here γ is Lebesgue measure on R.
Thus, r exists if some gω(t) is not too badly unbounded. This theorem is
stated for the class of “minimal” functions (see below), which includes the
a.p. ones.
