Let $d(x,y):=\min(x-y-\lfloor x-y\rfloor,y-x-\lfloor y-x\rfloor)$ (think about the geodesic distance of points on the circle), for each real number $a$ there is the monotone sequence $(\min_{0\le i<j<n}d(ia,ja))_n$. Is there some maximal sequence in this space, in the sense that every other sequence converges asymptotically faster or equally fast to $0$?
It seems that the sequence for the golden ratio $\phi$ converges very slowly, is this connected to the continued fraction property?
Yes. The points $x_k := ka-\lfloor ka \rfloor$ for $0 \le k \le n-1$ split $[0,1[$ into $k$ intervals, and you look at the shortest length. The three gap theorem states that the $k$ intervals have at most three different lengthes, and these lengthes can be computed explicitly with the help of the continued fraction expansion of $a$. View for example https://en.wikipedia.org/wiki/Three-gap_theorem and https://www.theoremoftheday.org/Docs/3dAlessandriBerthe.pdf
Golden ratio has the least partial quotients, and therefore the least decay of the lengthes.