New sum-product type estimates over finite fields

By Oliver Roche-Newton, M. Rudnev, Ilya D. Shkredov. 2014

88 citations recorded by the source corpus.

Let $F$ be a field with positive odd characteristic $p$. We prove a variety of new sum-product type estimates over $F$. They are derived from the theorem that the number of incidences between $m$ points and $n$ planes in the projective three-space $PG(3,F)$, with $m\geq n=O(p^2)$, is $$O( m\sqrt{n} + km ),$$ where $k$ denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that \begin{equation}\label{mres} |A\pm A|+|A\cdot A| =\Omega \left(|A|^{1+\frac{1}{5}}\right), \end{equation} for any $A$ such that $|A|

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/28f924b16a73677126a5692e18a70ef26c7d57d0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Discrete Mathematics and Combinatorics

Citations in the corpus

An improved point‐line incidence bound over arbitrary fields (2016, background): “If one replaces his Lemma 11 by a suitable multiplicative analogue of Theorem 6 of [17] (which holds over any field), it is a short calculation to show that his bound is improved to ǫ ≥ 1 326 .”
On the energy variant of the sum-product conjecture (2016, methodology): “The sum-product conjecture remains open although current world records in [17] for the reals and [20], [1] in positive characteristic have edged nearer to the statement of the conjecture.”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, background): “The sum-product estimates proven in [32] and [1] imply the following two statements for all A ⊂ F.”
Growth Estimates in Positive Characteristic via Collisions (2015, methodology): “The following Proposition 4 in the special case d = 1 was proven in [23].”