This problem is actually equivalent to the cap set problem, which asks for the largest subset of formula_148 such that no 3 points lie on a line. The cap set problem can be seen as a generalization of the card game Set. In 1982, Brown and Buhler were the first to show that formula_152 In 1995, Roy Mesuhlam used a similar technique to the Fourier-analytic proof of Roth's theorem to show that formula_153 This bound was improved to formula_154 in 2012 by Bateman and Katz. In 2016, Ernie Croot, Vsevolod Lev, Péter Pál Pach, Jordan Ellenberg and Dion Gijswijt developed a new technique based on the polynomial method to prove that formula_155. The best known lower bound is approximately formula_156, given in 2004 by Eden. Another generalization of Roth's Theorem shows that for positive density subsets, there not only exists a 3-term arithmetic progression, but that there exist many 3-APs all with the same common difference. Roth's theorem with popular differences: For all formula_104, there exists some formula_158 such that for every formula_159 and formula_160 with formula_161 there exists some formula_162 such that formula_163  If formula_19 is chosen randomly from formula_165 then we'd expect there to be formula_166 progressions for each value of formula_167. The popular differences theorem thus states that for each formula_24 with positive density, there is some formula_167 such that the number of 3-APs with common difference formula_167 is close to what we would expect. This theorem was first proven by Green in 2005, who gave a bound of formula_171 where formula_172 is the tower function. In 2019, Fox and Pham recently improved the bound to formula_173  A corresponding statement is also true in formula_174 for both 3-APs and 4-APs. However, the claim has been shown to be false for 5-APs.