\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=1.9]
 % coordinates (t,y) = (x/lambda, beta x/lambda - 1/4) mod 1 near (0,0), i.e. (Z,W)=(1,1);
 % near this point the sheet {p_a=0} is t + K_a y = 0, drawn as y = -s t with s = 1/K_a
 \begin{scope}
  \clip (-1.1,-1.6) rectangle (2.6,1.1);
  \draw[blue!70!black,thick] (-1.1,1.045) -- (2.6,-2.47);
  \draw[orange!85!black,thick] (-1.1,0.495) -- (2.6,-1.17);
  \draw[green!45!black,thick] (-1.1,0.132) -- (2.6,-0.312);
  % two passes of the line x -> (Z(x),W(x)), of slope beta (about 3.4)
  \draw[gray!55!black] ({0.5-1.6/3.4},-1.6) -- ({0.5+1.1/3.4},1.1);
  \draw[gray!55!black] ({1.9-1.6/3.4},-1.6) -- ({1.9+1.1/3.4},1.1);
 \end{scope}
 \draw[->,gray] (-1.1,0) -- (2.7,0) node[right,font=\small,black]{$t$};
 \draw[->,gray] (0,-1.6) -- (0,1.2) node[above,font=\small,black]{$y$};
 \fill (0,0) circle (1.6pt);
 \node[font=\scriptsize,anchor=north east] at (-0.06,-0.06) {$(Z,W)=(1,1)$};
 % distance eta from the common point to the near pass
 \draw[dashed] (0,0) -- (0.460,-0.135);
 \node[font=\scriptsize] at (0.17,-0.15) {$\eta$};
 % crossings: near pass (a tight cluster) and far pass (well separated)
 \foreach \p in {(0.391,-0.371),(0.442,-0.199),(0.483,-0.058)}
   \fill[red!80!black] \p circle (1.8pt);
 \foreach \p in {(1.485,-1.411),(1.678,-0.755),(1.835,-0.220)}
   \fill \p circle (1.8pt);
 \node[font=\scriptsize,anchor=east] at (-1.12,1.045) {$p_{a_1}=0$};
 \node[font=\scriptsize,anchor=east] at (-1.12,0.495) {$p_{a_2}=0$};
 \node[font=\scriptsize,anchor=east] at (-1.12,0.132) {$p_{a_3}=0$};
 \node[font=\scriptsize,anchor=south] at ({0.5+1.1/3.4},1.1) {$x\mapsto(Z(x),W(x))$};
 \node[font=\scriptsize,red!80!black,align=right,anchor=north east] (cl) at (-0.12,-0.6)
   {cluster of $s$ atoms within $O(\eta)$,\\ masses of size $\eta^{-(s-1)}$};
 \draw[->,red!80!black] (cl.east) -- (0.36,-0.40);
 \node[font=\scriptsize,align=left,anchor=west] at (2.0,-1.42)
   {separated atoms,\\ bounded masses};
\end{tikzpicture}
\caption{The mechanism of Theorem~\ref{thm:D} near the point $(Z,W)=(1,1)$, in the
coordinates $t=x/\lambda$ and $y=\beta x/\lambda-\frac14$ modulo $1$ (schematic,
$s=3$). All sheets $\{p_{a_j}=0\}$ pass through this point, with different slopes.
The real line is mapped to the line $x\mapsto(Z(x),W(x))$ of slope $\beta$, which
never meets the point but returns arbitrarily close to it. When it passes at
distance $\eta$, it crosses the $s$ sheets at $s$ atoms within distance $O(\eta)$ of
one another, and the masses, whose denominator $D_\beta P$ is of size $\eta^{s-1}$,
are of size $\eta^{-(s-1)}$. At these returns $\eta$ is of order $1/|x|$, from above
and from below, by the arithmetic of $\beta$ (Lemmas~\ref{lem:beta}
and~\ref{lem:sheets}).}\label{fig:sheets}
\end{figure}