We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.
In this paper, we show that as τ → √−1∞, any zero of the Lam´e function converges to either ∞ or a ﬁnite point p satisfying Rep = 1 2 and e2πip being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line Rez = 1 2. We also compute explicitly the eigenvalue setofthecorresponding L2[0,1] eigenvalueproblemforsuchKdVpotentials, thus extends Takemura’s works [26, 27]. Our main idea is to apply the classiﬁcation result for simply-periodic KdV potentials by Gesztesy, Unterkoﬂer and Weikard  and the Darboux transformation.