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Growth Estimates in Positive Characteristic via Collisions | | Esen Aksoy Yazici
; Brendan Murphy
; Misha Rudnev
; Ilya Shkredov
; | | Date: |
21 Dec 2015 | | Abstract: | Let $F$ be a field of characteristic $p>2$ and $Asubset F$ have sufficiently
small cardinality in terms of $p$. We improve the state of the art of a variety
of sum-product type inequalities. In particular, we prove that $$ |AA|^2|A+A|^3
gg |A|^6,qquad |A(A+A)|gg |A|^{3/2}. $$ We also prove several two-variable
extractor estimates: ${displaystyle |A(A+1)| gg|A|^{9/8},}$ $$ |A+A^2|gg
|A|^{11/10},; |A+A^3|gg |A|^{29/28}, ; |A+1/A|gg |A|^{31/30}.$$
Besides, we address questions of cardinalities $|A+A|$ vs $|f(A)+f(A)|$, for
a polynomial $f$, where we establish the inequalities $$ max(|A+A|,,
|A^2+A^2|)gg |A|^{8/7}, ;; max(|A-A|,, |A^3+A^3|)gg |A|^{17/16}. $$
Szemer’edi-Trotter type implications of the arithmetic estimates in question
are that a Cartesian product point set $P=A imes B$ in $F^2$, of $n$ elements,
with $|B|leq |A|< p^{2/3}$ makes $O(n^{3/4}m^{2/3} + m + n)$ incidences with
any set of $m$ lines. In particular, when $|A|=|B|$, there are $ll n^{9/4}$
collinear triples of points in $P$, $gg n^{3/2}$ distinct lines between pairs
of its points, in $gg n^{3/4}$ distinct directions.
Besides, $P=A imes A$ determines $gg n^{9/16}$ distinct pair-wise
distances.
These estimates are obtained on the basis of a new plane geometry
interpretation of the incidence theorem between points and planes in three
dimensions, which we call collisions of images. | | Source: | arXiv, 1512.6613 | | Services: | Forum | Review | PDF | Favorites | |
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