Expanders with superquadratic growth

We prove several expanders with exponent strictly greater than 2. For any finite set A ⊂ ℝ, we prove the following six-variable expander results: (Formula Presented)

Four-variable expanders over the prime fields

Let $\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\mathbb{F}_p$ with very small size in terms of $p$. In this note, we show that the number of distinct cubic distances determined by points in $A\times A$ satisfies $$|(A-A)^3+(A-A)^3|\gg |A|^{8/7},$$ which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that $$\max \left\lbrace |A+A|, |f(A, A)|\right\rbrace\gg |A|^{6/5},$$ where $f(x, y)$ is a quadratic polynomial in $\mathbb{F}_p[x, y]$ that is not of the form $g(\alpha x+\beta y)$ for some univariate polynomial $g$.Comment: Accepted in PAMS, 201

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 Distinct Angles in the Plane

We prove that if $N$ points lie in convex position in the plane then they determine $N^{1+ 3/23-o(1)}$ distinct angles, provided the points do not lie on a common circle. This is the first super-linear bound on the distinct angles problem that has received recent attention. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=\Omega(N^{1/4})$ or $\Omega(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position, recently used by Solymosi. Convex position also allows us to divide our set into two large ordered pieces. We use the squeezing lemma and the level sets of repeated angles to estimate the number of angles formed between these pieces. We obtain the main theorem using Stevens and Warren's convexity versus sum-set bound.Comment: 23 pages, 10 figure