* lisp/progmodes/sh-script.el (sh--inside-arithmetic-expression): New func.

(sh-font-lock-open-heredoc): Use it.

Fixes: debbugs:12770

lisp/progmodes/sh-script.el

@@ -940,6 +940,15 @@ See `sh-feature'.")
     (concat "<<-?\\s-*\\\\?\\(\\(?:['\"][^'\"]+['\"]\\|\\sw\\|[-/~._]\\)+\\)"
             sh-escaped-line-re "\\(\n\\)")))
 
+(defun sh--inside-arithmetic-expression (pos)
+  (save-excursion
+    (let ((ppss (syntax-ppss pos)))
+      (when (nth 1 ppss)
+        (goto-char (nth 1 ppss))
+        (and (eq ?\( (char-after))
+             (eq ?\( (char-before))
+             (eq ?\$ (char-before (1- (point)))))))))
+
 (defun sh-font-lock-open-heredoc (start string eol)
   "Determine the syntax of the \\n after a <<EOF.
 START is the position of <<.
@@ -948,7 +957,8 @@ INDENTED is non-nil if the here document's content (and the EOF mark) can
 be indented (i.e. a <<- was used rather than just <<).
 Point is at the beginning of the next line."
   (unless (or (memq (char-before start) '(?< ?>))
-	      (sh-in-comment-or-string start))
+	      (sh-in-comment-or-string start)
+              (sh--inside-arithmetic-expression start))
     ;; We're looking at <<STRING, so we add "^STRING$" to the syntactic
     ;; font-lock keywords to detect the end of this here document.
     (let ((str (replace-regexp-in-string "['\"]" "" string))