I'm not sure how recent "contemporary" mathematics means so I'll mention a few things, some somewhat classical but all connected to the fact that $\phi=\frac{1+\sqrt{5}}{2}$ is closely connected to the Fibonacci sequence. I will endeavor to order these roughly in oldest to most recent.

First, in a certain sense, $\phi$ is the hardest number to approximate with rational numbers. What do we mean by that? if you have an irrational number $\alpha$, and you want to approximate $\alpha$ with rational numbers of the form $\frac{n}{d}$, then you can get approximations as good as you want by making $d$ larger. However, suppose you are interested in getting as close as you can and wanting to know how bad a price you pay in terms of increasing $d$, it turns out that by multiple ways of making this precise, $\phi$ is the worst possible. This is closely related to the fact that it has continued fraction $[1,1,1,1...]$ and so the best possible approximates actually have numerator and denominator Fibonacci numbers.

Understanding the Fibonacci numbers better turns out to be closely connected to the Binet formula, which says that
$$F_n = \frac{\left(\frac{1+\sqrt{5}}{2}\right)^n- \left(\frac{1-\sqrt{5}}{2} \right)^n}{\sqrt{5}}.$$

So for example, there's an old result that the Fibonacci sequence distributes over gcd, that is $$F_{\mathrm{gcd}(a,b)}= \mathrm{gcd}(F_a,F_b).$$ It turns out that one of the more enlightening ways of proving this is by using the Binet formula and then looking at how the Fibonacci numbers behave in the ring $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$.

This leads us to the more recent work. Define the order of apparition of $n$, denoted by $z(n)$, as the least $k$ such that $n∣F_k$. The last few years have seen extensive work on trying to understand this function, and the closely related function of the Pisano period, which says how long it takes $F_n$ to repeat mod $m$ for some $m$. In the early 2010s a whole bunch of papers on this topic were written by Diego Marques which are of note in this regard. One of the techniques here involves trying to understand $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$ really closely.

In a related note, there's a recent paper in the Journal of Number Theory by Roswitha Hofer which generalizes the golden ration to fields of formal power series. Hofer's paper can probably be used as a jumping off point for generalizing some of the results mentioned above to other contexts.

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