Large time behavior of a linear difference equation with rationally non-related delays

We consider the linear difference equation x ( t ) = ∑ j = 1 N a j x ( t − r j ) , t ≥ 0 , where a j > 0 , 1 ≤ j ≤ N , and the delays 0 < r 1 < r 2 < ⋯ < r N are not rationally related in the sense that r j / r k is irrational for some j and  k . It is shown that the large time behavior of the continuous solutions can be described in terms of the unique real root of the associated characteristic equation. The proof is based on Newman’s Tauberian theorem.