Hardness of Max-2Lin and Max-3Lin over integers, reals, and large cyclic groups

In 1997, Håstad showed NP-hardness of (1 - ε, 1/q + δ)-approximating Max-3Lin(Z_q); however, it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of (1 - ε, δ)-approximating Max-3Lin(Z). In 2004, Khot-Kindler-Mossel-O'Donnell showed UG-hardness of (1 - ε, δ)-approximating Max-2Lin(Z_q) for q = q(ε,δ) a sufficiently large constant; however, achieving the same hardness for Max-2Lin(Z) was given as an open problem in Raghavendra's 2009 thesis. In this work, we show that fairly simple modifications to the proofs of the Max-3Lin(Z_q) and Max-2Lin(Z_q) results yield optimal hardness results over Z. In fact, we show a kind of "bicriteria" hardness: Even when there is a (1 - ε)-good solution over Z, it is hard for an algorithm to find a δ-good solution over Z, R, or Z_m for any m ≥ q(ε, δ) of the algorithm's choosing.