Let $\Gamma$ be a finite directed graph, and suppose each directed edge $e \colon a \to b$ has a positive real length. Suppose given vertices $x, y \in \Gamma$, and suppose there are infinitely many directed walks from $x$ to $y$. For any $\varepsilon > 0$, I am interested in finding $N \in \mathbb{R}$ so that for every $L > N$, there exists a walk from $x$ to $y$ whose length is within $\varepsilon$ of $L$.
Of course, such an $N$ may not exist. For example, if every edge length is an integer, then we can never accommodate $\varepsilon < 1/2$. However, $x$ admits two self-walks with lengths that are linearly independent over $\mathbb{Q}$, then it seems that $N = N(\Gamma, \varepsilon)$ always exists.
How can I find a bound on $N$ as a function of $\Gamma$ and $\varepsilon$?
