\begin{figure}[t]
\centering
\begin{tikzpicture}[x=1.30cm,y=1cm]
 % (a) one block, x >= 0 (the set is symmetric); R=3, k=5, schematic distances
 \fill[gray!15] (0,-0.02) rectangle (2.5,0.42);
 \node[font=\scriptsize] at (1.25,0.2) {no atoms};
 \draw[->] (0,0) -- (10.75,0) node[right,font=\small]{$x$};
 \foreach \n in {0,...,10} \draw (\n,0) -- (\n,-0.08) node[below,font=\scriptsize]{$\n$};
 \foreach \n in {3,...,10} {
   \foreach \b in {0.05,0.15,0.25,0.35,0.45} {
     \draw[line width=0.6pt] (\n-\b,0) -- (\n-\b,0.22);
     \draw[line width=0.6pt] (\n+\b,0) -- (\n+\b,0.22);
   }
 }
 \foreach \n in {6,...,10} {
   \pgfmathtruncatemacro{\J}{\n-5}
   \foreach \j in {1,...,\J} {
     \pgfmathsetmacro{\e}{0.035*0.5^(\j-1)}
     \draw[red!80!black,line width=0.6pt] (\n+0.5-\e,0) -- (\n+0.5-\e,0.36);
     \draw[red!80!black,line width=0.6pt] (\n-0.5+\e,0) -- (\n-0.5+\e,0.36);
   }
 }
 \draw[decorate,decoration={brace,amplitude=4pt}] (2.5,0.5) -- (10.5,0.5)
   node[pos=0.2,above=5pt,font=\scriptsize]{head atoms $n\pm\beta_j$ in all cells $|n|\ge R$};
 \draw[decorate,decoration={brace,amplitude=4pt},red!80!black] (5.5,1.45) -- (10.5,1.45)
   node[midway,above=5pt,font=\scriptsize]{tail atoms $n\pm\gamma_j$ in the cells $|n|\ge k+j$, piling up at the half-integers};
 \node[font=\small] at (-0.3,0.25) {(a)};
\end{tikzpicture}

\medskip
\begin{tikzpicture}[x=25cm,y=1cm]
 % (b) zoom near the half-integer 9 + 1/2
 \draw[->] (9.22,0) -- (9.78,0) node[right,font=\small]{$x$};
 \draw[dashed,gray] (9.5,-0.15) -- (9.5,0.62);
 \node[below,font=\scriptsize] at (9.5,-0.12) {$n+\frac12$};
 \foreach \x/\l in {9.25/{n+\beta_3},9.35/{n+\beta_4},9.45/{n+\beta_5},9.55/{n+1-\beta_5},9.65/{n+1-\beta_4},9.75/{n+1-\beta_3}}
   {\draw[line width=0.8pt] (\x,0) -- (\x,0.3); \node[below,font=\scriptsize] at (\x,-0.05) {$\l$};}
 \foreach \j in {1,...,4} {
   \pgfmathsetmacro{\e}{0.035*0.5^(\j-1)}
   \draw[red!80!black,line width=0.8pt] (9.5-\e,0) -- (9.5-\e,0.45);
 }
 \foreach \j in {1,...,5} {
   \pgfmathsetmacro{\e}{0.035*0.5^(\j-1)}
   \draw[red!80!black,line width=0.8pt] (9.5+\e,0) -- (9.5+\e,0.45);
 }
 \node[font=\scriptsize,red!80!black,align=center] at (9.5,0.95)
   {$n+\gamma_j=n+\frac12-\eps_j$ and $n+1-\gamma_j=n+\frac12+\eps_j$};
 \node[font=\small] at (9.2,0.3) {(b)};
\end{tikzpicture}
\caption{One block $C(P,R)$ of Section~\ref{sec:blocks}, schematically, with
$R=3$, $k=2R-1=5$, and only $x\ge0$ shown (the set is symmetric). (a)~There are
no atoms in the central interval. The head phases $\beta_j=(2j-1)/(4k)$ occur in
all cells $|n|\ge R$; the tail phases $\gamma_j=\frac12-\eps_j$ occur in the cells
$|n|\ge k+j$, so that farther cells carry more of them. (b)~Near a half-integer
$n+\frac12$, the tail atoms of the two adjacent cells form a cluster that tightens
from both sides. The distances $\eps_j$ are greatly exaggerated: in the
construction they decrease doubly exponentially. Such clusters play the role of
the derivatives of a Dirac mass at $n+\frac12$ in the heuristic \eqref{eq:toy}.}
\label{fig:block}
\end{figure}