Linearly bounded liars, adaptive covering codes, and deterministic random walks

We analyze a deterministic form of the random walk on the integer line called the liar machine, similar to the rotor-router model, finding asymptotically tight pointwise and interval discrepancy bounds versus random walk. This provides an improvement in the best-known winning strategies in the binary symmetric pathological liar game with a linear fraction of responses allowed to be lies. Equivalently, this proves the existence of adaptive binary block covering codes with block length n, covering radius ≤ fn for f ∈ (0, 1/2), and cardinality O( √ log log n/(1−2f)) times the sphere bound 2/ ( n ≤bfnc ) .