Almost Periodic Flows and Solutions of Differential Equations

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 A ν (m) = exp ( i ∑ ν = 1 J ( ν )r j λ jτ ′ j) , and similarly exp ( i ∑ ν = 1 J ( ν )r j λ jτ ′′ 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 q 1 (v)(l) ⩽ J for all v, l, j . For if J = 2, P l ( v ) = 1, P 2( v ) = v – 1, q 1 ( v ) = q 2 ( v ) = v , we have1v+v ‐ 1v= 0 ( mod 1 ) ,while q 1 ( v ) and q 2 ( 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 p j (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 p j (l), l = 1, 2, …, J in P , and adding, we obtain P p j ( v )/q j ( v = 0 (mod 1), j = 1, 2, …, J, v = 1, 2, …. Since P j ( v ), q j ( v ) are prime to one another, q j ( 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 Q j = max q j ( v ), v = 1, 2, …, and then γ j = χ j Q j ! is an integral base. For ( P j ( v )Q j !)/q j ( v ) is an integer for all v and j .