2016/12/25 by Ge, Gennian, Shangguan, Chong
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1612.08255
Recently, Croot, Lev, and Pach (Ann. of Math., 185:331--337, 2017.) and Ellenberg and Gijswijt (Ann. of Math., 185:339--443, 2017.) developed a new polynomial method and used it to prove upper bounds for three-term arithmetic progression free sets in ℤ4n and \mathbbF3n, respectively. Their approach was later summarized by Tao and is now known as the slice rank method. In this paper, we apply this method to obtain a new upper bound on the cardinality of subsets of \mathbbFnq which contain no right angles. More precisely, let q be a fixed odd prime power and x⋅ y be the standard inner product of two vectors x,y∈\mathbbFqn, we prove that the maximum cardinality of a subset A⊆\mathbbFqn without three distinct elements x,y,z∈ A satisfying (z-x)⋅ (y-x)=0 is at most \binomn+qq-1+3. For sufficiently large n, our result significantly improves the previous upper bound of Bennett (European J. Combin., 70:155--163, 2018.), who showed that |A|=O(q(n+2)/(3)).