On distinct cross-ratios and related growth problems

It is shown that for a finite set $A$ of four or more complex numbers, the cardinality of the set $C[A]$ of all cross-ratios generated by quadruples of pair-wise distinct elements of $A$ is $|C[A]|\gg |A|^{2+\frac{2}{11}}\log^{-\frac{6}{11}} |A|$ and without the logarithmic factor in the real case. The set $C=C[A]$ always grows under both addition and multiplication. The cross-ratio arises, in particular, in the study of the open question of the minimum number of triangle areas, with two vertices in a given non-collinear finite point set in the plane and the third one at the fixed origin. The above distinct cross-ratio bound implies a new lower bound for the latter question, and enables one to show growth of the set $\sin(A-A),\;A\subset \mathbb R/\pi\mathbb Z$ under multiplication. It seems reasonable to conjecture that more-fold product, as well as sum sets of this set or $C$ continue growing ad infinitum.Comment: 9p

On distance measures for well-distributed sets

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions, we link this problem to the question about the existence of smooth well-curved hypersurfaces that support many integer points

An explicit incidence theorem in F_p

Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over \F_p ($n<\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant.Comment: 11 page