Chacon eigenfunctions are constant

By F1, $\mathrm{\mid }\lambda \mathrm{\mid }=1$, so $\mathrm{\mid }f\mathrm{\mid }\circ T=\mathrm{\mid }f\mathrm{\mid }$ a.e. Choose a finite-valued measurable representative of the $L^2$ class by setting it to zero on its null exceptional set. F2–F5 make $\mathrm{\mid }f\mathrm{\mid }$ a constant $c$ a.e. Nonzeroness forces $c>0$. Divide by $c$, so henceforth $\mathrm{\mid }f\mathrm{\mid }=1$ a.e. For each positive integer $k$, iteration gives $f(T^{k}x)={\lambda }^{k}f(x)$ outside the finite union of preimages of the original exceptional null set. F4's measure preservation makes that union null. These relations may therefore be used for either finite return time below.

Fix $\epsilon >0$ and $0<\delta <1\mathrm{/}6$. Cover the unit circle by finitely many open disks of radius $\epsilon$ with centers on the circle: equally spaced arguments with spacing less than $\epsilon$ suffice, using $\mathrm{\mid }{e}^{it}-e^{is}\mathrm{\mid }\le \mathrm{\mid }t-s\mathrm{\mid }$. Since $\mathrm{\mid }f\mathrm{\mid }=1$ a.e., at least one disk centered at $a$, $\mathrm{\mid }a\mathrm{\mid }=1$, has positive-measure inverse image $E=\{x:\mathrm{\mid }f(x)-a\mathrm{\mid }<\epsilon \}$. By F3 choose a level $J=L_{r,j}$ with $\mu (J\setminus E)<\delta w_r$. F7's next-stage ordering places $J^{(1)}$ exactly $h_r$ levels after $J^{(0)}$ and $J^{(2)}$ exactly $h_r+1$ levels after $J^{(1)}$, because the latter route crosses the one spacer. Together with F4 this gives $T^{h_r}:J^{(0)}\to J^{(1)}$ and $T^{h_r+1}:J^{(1)}\to J^{(2)}$ as measure-preserving translations on the invariant conull set.

For the first route, the set of points $x\in J^{(0)}$ for which either $x\mathrm{\notin }E$ or $T^{h_r}x\mathrm{\notin }E$ has measure at most $2\delta w_r$. Thus a set of measure at least $(1\mathrm{/}3-2\delta )w_r>0$ satisfies both memberships and the eigenfunction iterate relation. At one such point, $\mathrm{\mid }{\lambda }^{h_r}a-a\mathrm{\mid }\le \mathrm{\mid }{\lambda }^{h_r}(a-f(x))\mathrm{\mid }+\mathrm{\mid }f(T^{h_r}x)-a\mathrm{\mid }<2\epsilon$. The same argument on $J^{(1)}$ with return time $h_r+1$ gives $\mathrm{\mid }{\lambda }^{h_r+1}-1\mathrm{\mid }<2\epsilon$. Null exceptions from step 1.1 and the conull tower convention do not change positive measure.

Since $\mathrm{\mid }\lambda \mathrm{\mid }=1$, $\mathrm{\mid }\lambda -1\mathrm{\mid }=\mathrm{\mid }{\lambda }^{h_r+1}-{\lambda }^{h_r}\mathrm{\mid }\le \mathrm{\mid }{\lambda }^{h_r+1}-1\mathrm{\mid }+\mathrm{\mid }{\lambda }^{h_r}-1\mathrm{\mid }<4\epsilon$. Every positive $\epsilon$ is allowed, so $\lambda =1$. Now F5 makes $f$ constant a.e., and undoing the normalization preserves constancy. AC is inherited from the tower and ergodicity inputs; the disk and positive-measure witnesses require only finite choices for each fixed epsilon.