Running least common multiples: distinct values and multiplicities

Reversals near the origin.

A shorter route would take, at a return of the orbit near the origin of $T_0$, the chain of crossings there. That shortcut need not work when $\dim T_0=2$, which happens exactly when the numbers $1/\log q$ span a two-dimensional space over $\mathbb{Q}$; the proof makes no assumption excluding this case. Linear-form estimates for logarithms do not supply the required independence of their reciprocals. In that case every small vector of $L$, in the coordinates ${\theta }_q$, has the form $\alpha (1,\dots ,1)+\beta w$ for one fixed $w$, so near the origin all the forms ${\theta }_b-{\theta }_a$ are proportional. The chain at such a return contains every prime of $P$, and the only rearrangement it can undergo is the full reversal. Full reversals can collide:

and among the sets of four primes below $100$ there are five such collisions. Remark 5.5 avoids the question by producing a point on exactly two walls for every $P$ with $|P|\ge 4$; a point on three walls would also serve, by Remark 5.4. Swapping one adjacent pair always changes $f$, since $f(\sigma )-f({\sigma }^{\prime })=({\sigma }_{i+1}-{\sigma }_i){\sigma }_{i+2}\cdots {\sigma }_n$ when ${\sigma }^{\prime }$ swaps ${\sigma }_i$ and ${\sigma }_{i+1}$; a rearranged chain of four or more primes can differ from its image by more than one swap. Nor can the argument stop at the first rearranged chain: for four primes that chain can collide, as $f(5,7,3,2)=f(7,2,3,5)=51$ shows, which is why Remark 5.3(i) is applied to every chain.