The Density Finite Sums Theorem

The uploaded paper proves the density finite sums theorem up to any fixed length k with a common shift (Theorem 1.1), and reduces the combinatorial statement to a dynamical one (Theorem 1.2), then back via an Erdős-progression-to-finite-sums lemma and a tailored correspondence principle. The key statements and the flow Theorem 1.2 ⇒ Lemma 2.2 ⇒ Theorem 1.1 are explicitly given in the manuscript. The candidate solution outlines exactly this route: Furstenberg correspondence; apply the KMRR dynamical result; then translate back. Minor slips: it attributes the “finite sums hit after a shift” conclusion directly to Theorem 1.2, whereas in the paper this conclusion comes from combining Theorem 1.2 with Lemma 2.2; and it cites a more standard inequality-style correspondence principle rather than the paper’s stronger coding version. These do not affect correctness of the argument. See Theorem 1.1 (density finite sums with shift) and its proof-by-reduction to Theorem 1.2, together with Lemma 2.2 and Theorem 2.3 in the paper .