While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, t_n$ satisfies $$ N(t) = N(t-t_1) + N(t-t_2) + \dots + N(t-t_n). \quad \quad \quad (1)$$
Then, he says, "according to a well-known result in finite differences", $N(t)$ is asymptotic for large $t$ to $X_0^t$ where $X_0$ is the largest real solution of the characteristic equation $$ X^{-t_1} + X^{-t_2} + \dots + X^{-t_n} = 1. \quad \quad \quad (2)$$
My question: what is this well-known theory, and where can I read about it? (If the proof is short enough to outline here in the space of a math.SE answer that would be great too, of course!) What sort of an equation is $(2)$, anyway?
[Some fuzzy attempts follow.]
I could not prove this, but I can non-rigorously find the result plausible: for instance, if we guess that $N(t)$ is of the form $X^t$ (or even $c X^t$) for some $X$, then plugging in $N(t) = c X^t$ into the equation $(1)$ gives $$ cX^{t} = cX^{t-t_1} + cX^{t-t_2} + \dots + cX^{t-t_n}$$ so dividing by $cX^t$ we get equation $(2)$, $$ 1 = X^{-t_1} + X^{-t_2} + \dots + X^{-t_n}$$ And of course for any solution $X$ of the above equation and for any constant $c$, we can take $N(t) = cX^t$. Also linear combinations of solutions to $(1)$ are also solutions, so $N(t)$ could even be of the form $$N(t) = c_0X_0^t + c_1X_1^t + \dots$$ in which asymptotically only the largest solution $X_0$ matters. That is, $$\lim_{t\to\infty}\frac{N(t)}{X_0^t} = \lim_{t\to\infty}\left(c_0 + \frac{c_1X_1^t}{X_0^t} + \dots \right)= c_0.$$
Now if were working with recurrence relations over the integers — $N\colon \mathbb{Z}\to\mathbb{R}$, say, with all the $t_i$ being integers, especially if they are integers $1$ to $n$ — then I guess we could further say that we have found an $n$-dimensional space of solutions (here $n$ being the number of solutions to equation $(2)$), and the solution space also is of dimension $n$ (it has $n$ "degrees of freedom" because the first $n$ values determine the sequence — all this needs to be made more rigorous and to consider more general $t_i$), and therefore these must be all the solutions. That would complete the "proof".
But in this real-number case I'm completely clueless how one would prove a thing like this, and everything I wrote above may be entirely the wrong tack to pursue.
This assertion is certainly false as stated. For one thing, you certainly need some "regularity" assumptions, otherwise e.g. there are solutions of $N(t) = N(t+1)$ that are unbounded on each interval of length 1, and so don't satisfy any asymptotics. But even "nice" solutions might involve non-real solutions of the characteristic equation. For example, $N(t) = c^t \sin(\pi t)$ satisfies $N(t) = N(t-1) + N(t+2)$ where $c$ is the real root of $c^3 - c - 1$, approximately $1.324717957$, and this is certainly not asymptotic to any $X^t$.