Sum–product estimates for rational functions

@article{Bukh2012SumproductEF,
  title={Sum–product estimates for rational functions},
  author={Boris Bukh and Jacob Tsimerman},
  journal={Proceedings of the London Mathematical Society},
  year={2012},
  volume={104}
}
We establish several sum–product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)+|AA| is substantially larger than |A| for an arbitrary polynomial f over 𝔽p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)+|g(A, A)| can be as small as |A| for large |A|. Third, we show that under mild conditions on f, |f(A, A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what… 
Conditional expanding bounds for two-variable functions over prime fields
Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances
TLDR
It is proved that for a quadratic polynomial f ∈ F, which is not of the form g(h(x)+k(y)+l(z), the authors have |f(A×B×C)|≫N3/2 for any sets A,B,C ⊂ $$\mathbb{F}^d$$Fd determines almost |A|2 distinct distances if |A | is not too large.
Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets
  • T. Tao
  • Mathematics
    Contributions Discret. Math.
  • 2015
TLDR
An algebraic regularity lemma is established that describes the structure of dense graphs generated by definable subsets over finite fields of large characteristic.
Semialgebraic methods and generalized sum-product phenomena
For a bivariate P (x, y) ∈ R[x, y] \ (R[x] ∪R[y]), our first result shows that for all finite A ⊆ R, |P (A,A)| ≥ α|A| with α = α(degP ) ∈ R unless P (x, y) = f(γu(x) + δu(y)) or P (x, y) =
Partition Rank and Analytic Rank are Uniformly Equivalent
We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality, independently of the number of
Growth Estimates in Positive Characteristic via Collisions
Let $F$ be a field of characteristic $p>2$ and $A\subset F$ have sufficiently small cardinality in terms of $p$. We improve the state of the art of a variety of sum-product type inequalities. In
On the concentration of points of polynomial maps and applications
For a polynomial $${f\in{\mathbb {F}}_p[X]}$$ , we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in
Polynomial Values in Subfields and Affine Subspaces of Finite Fields
For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by
...
...

References

SHOWING 1-10 OF 32 REFERENCES
SUM-PRODUCT ESTIMATES VIA DIRECTED EXPANDERS
Let Fq be a finite field of order q and P be a polynomial in Fq [x1, x2]. For a set A ⊂ Fq , define P (A) := {P (x1, x2)|xi ∈ A}. Using certain constructions of expanders, we characterize all
On a variant of sum-product estimates and explicit exponential sum bounds in prime fields
  • J. Bourgain, M. Garaev
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2009
Abstract Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound $$
THE SUM-PRODUCT ESTIMATE FOR LARGE SUBSETS OF PRIME FIELDS
Let Fp be the field of prime order p. It is known that for any integer N ∈ [1,p] one can construct a subset A C Fp with |A| = N such that max{|A + A|, |AA|} «p 1/2 |A| 1/2 . One of the results of the
MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS
In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when
A sum-product estimate in finite fields, and applications
TLDR
A Szemerédi-Trotter type theorem in finite fields is proved, and a new estimate for the Erdös distance problem in finite field, as well as the three-dimensional Kakeya problem in infinite fields is obtained.
BOUNDS ON ARITHMETIC PROJECTIONS, AND APPLICATIONS TO THE KAKEYA CONJECTURE
Let A, B, be finite subsets of a torsion-free abelian group, and let G ⊂ A × B be suchth at # A, #B,#{a + b :( a, b) ∈ G }≤ N. We consider the question of estimating the quantity #{a − b :( a, b) ∈
Fourier analysis and expanding phenomena in finite fields
In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In
Mapping incidences
We show that any finite set S in a characteristic zero integral domain can be mapped to ℤ/pℤ, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a
On a question of Erdős and Moser
For two finite sets of real numbers A and B, one says that B is sum-free with respect to A if the sum set {b + b | b, b ∈ B, b 6= b} is disjoint from A. Forty years ago, Erdős and Moser posed the
...
...