KöMaL 2026 A935

A criminal starts at the origin and escapes by moving on the integer lattice points of the plane. At each step, they move to a neighboring lattice point, with the restriction that they never move in the same direction twice consecutively. It is known that their first move is upward, and that their path visits every lattice point exactly once. An agent also starts at the origin but cannot see the criminal. Fortunately, the headquarters can track the criminal's movement, but they can communicate with the agent only in the following way: a predetermined infinite set $S \subseteq \mathbb{N}$ is given, and for each $s \in S$, after the $s$-th step of the criminal, a message consisting of either $1$ or $2$ is sent to the agent. The agent moves only when a message is received, and always makes exactly as many steps as the criminal has taken since the last message. What is the value of $I=\inf_{S} \left(\lim_{n\rightarrow \infty} \frac{|S \cap \{1,2,3,\ldots,n\}|}{n} \right)$ taken over all sets $S$ for which the agent can uniquely reconstruct the movement of the criminal? Both the agent and the headquarters are aware of the rules restricting the criminal's movement.