Almost Periodic Flows and Solutions of Differential Equations
M. L. Cartwright · Proceedings of the London Mathematical Society · 1967
1. In the proof of Theorem 7 on p. 365 the method of Theorem 6 shows that ϕ(t,m) ∼ ∑ ν = 1 ∞ A ν ( m ) exp ( i Λ ν t ) where Aν(m) = exp ( i ∑ ν = 1 J ( ν ) r j λ j τ ′ j ) , and similarly exp ( i ∑ ν = 1 J ( ν ) r j λ j τ ′′ j ) This gives (8) more directly than the procedure indicated. 2. In the proof of Theorem 10 on p. 370 it does not follow from (3) that q1(v)(l) ⩽ J for all v, l, j. For if J = 2, Pl(v) = 1, P2(v) = v – 1, q1(v) = q2(v) = v, we have 1 v + v - 1 v = 0 ( mod 1 ) , while q1(v) and q2(v) tend to ∞ with v. However, the points in the τ space τ ( l ) = ( 2 π p 1 ( l ) λ 1 , 2 π p 2 ( l ) λ 2 , … , 2 π p J ( l ) λ J ) , l = 1 , 2 , … , J , are linearly independent, and so P = det pj(l) ⊆ 0. For each fixed v, multiplying the equations (3) on p. 370, viz. ∑ j - 1 J ( v ) p j ( l ) q j ( l ) , P j ( l ) = 0 ( mod 1 ) , l = 1 , 2 , … , J , by the cofactors of pj(l), l = 1, 2, …, J in P, and adding, we obtain Ppj(v)/qj(v = 0 (mod 1), j = 1, 2, …, J, v = 1, 2, …. Since Pj(v), qj(v) are prime to one another, qj(v) is a factor of P, and since P is independent of v, |j(v)| ⩽ P for all j and v. Hence we may put Qj = max qj(v), v = 1, 2, …, and then γj = χjQj! is an integral base. For (Pj(v)Qj!)/qj(v) is an integer for all v and j.