Abstract. In this paper, we strengthen a result by Green about an analogue of Sárközy’s theorem in the setting of polynomial rings [Formula: see text]. In the integer setting, for a given polynomial [Formula: see text] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset [Formula: see text] that does not contain distinct [Formula: see text] satisfying [Formula: see text] for some [Formula: see text]. Green proved an analogous result with much stronger bounds in the setting of subsets [Formula: see text] of the polynomial ring [Formula: see text], but this result required the additional condition that the number of roots of the polynomial [Formula: see text] be coprime to [Formula: see text]. We generalize Green’s result, removing this condition. As an application, we also obtain a version of Sárközy’s theorem with similar strong bounds for subsets [Formula: see text] for [Formula: see text] for a fixed prime [Formula: see text] and large [Formula: see text].

List-decodability of Reed-Solomon codes has re-ceived a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form$r=1-\varepsilon$for$\varepsilon$tending to zero. Our main result states that there exist Reed-Solomon codes with rate$\Omega(\varepsilon)$which are$(1-\varepsilon, O(1/\varepsilon)$-list-decodable, meaning that any Hamming ball of radius$1-\varepsilon$contains at most$O(1/\varepsilon)$codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance.