Abstract
In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is δ>0 such that for any P1,⋯,P4, and Q1,⋯,Qn∈C2, if there are ≤n21+δ many distinct lines between Pi and Qj for all i,j, then P1,⋯,P4 are collinear. If the number of the distinct lines is thenthecrossratioofthefourpointsisalgebraic.\endproclaim\proclaim2Given c>0,thereis \delta 0 suchthatforany P_1, P_2, P_3 noncollinear,and Q_1, \cdots, Q_n \in \Bbb C^2,ifthereare \leq c n^{1/2} manydistinctlinesbetween P_i and Q_j forall i,j,thenforany P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\},wehave \delta n distinctlinesbetween P and Q_j.\endproclaim\proclaim3Given c>0,thereis \epsilon 0 suchthatforany P_1, P_2, P_3 collinear,and Q_1, \cdots, Q_n \in \Bbb C^2 (respectively, \Bbb R^2),ifthereare \leq c n^{1/2} manydistinctlinesbetween P_i and Q_j forall i,j,thenforany P notlyingontheline L(P_1,P_2),wehaveatleast n^{1-\epsilon} (resp. n/\log n)distinctlinesbetween P and Q_j.\endproclaimThemainingredientsusedarethesubspacetheorem,Balog−Szemereˊdi−GowersTheorem,andSzemereˊdi−TrotterTheorem.Wealsogeneralizethetheoremstohighdimensions,extendTheorem1to \Bbb F_p^2,andgivetheversionofTheorem2over \Bbb Q$.