Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012]. In contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes. The finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].

We bound separations between the entangled and classical values for several classes of nonlocal $t$-player games. Our motivating question is whether there is a family of $t$-player XOR games for which the entangled bias is $1$ but for which the classical bias goes down to $0$, for fixed $t$. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-$m$ games and show that their classical value is always at least $\frac{1}{m} + \frac{m-1}{m} t^{1-t}$. Secondly, for free XOR games, in which the input distribution is of product form, we show $β(G) \geq β^*(G)^{2^t}$ where $β(G)$ and $β^*(G)$ are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is $1-ε$ then the classical value is at least $1-\mathcal{O}(\sqrt{ε\log k})$ where $k$ is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.