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.