23 research outputs found
Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets
Let P: \F \times \F \to \F be a polynomial of bounded degree over a finite field \F of large characteristic. In this paper we establish the following dichotomy: either is a \emph{moderate asymmetric expander} in the sense that |P(A,B)| \gg |\F| whenever A, B \subset \F are such that |A| |B| \geq C |\F|^{2-1/8} for a sufficiently large , or else takes the form or for some polynomials . This is a reasonably satisfactory classification of polynomials of two variables that moderately expand (either symmetrically or asymmetrically). We obtain a similar classification for weak expansion (in which one has |P(A,A)| \gg |A|^{1/2} |\F|^{1/2} whenever |A| \geq C |\F|^{1-1/16}), and a partially satisfactory classification for almost strong asymmetric expansion (in which |P(A,B)| = (1-O(|\F|^{-c})) |\F| when |A|, |B| \geq |\F|^{1-c} for some small absolute constant c>0). The main new tool used to establish these results is an \emph{algebraic regularity lemma} that describes the structure of dense graphs generated by definable subsets over finite fields of large characteristic. This lemma strengthens the Sz\'emeredi regularity lemma in the algebraic case, in that while the latter lemma decomposes a graph into a bounded number of components, most of which are \eps-regular for some small but fixed , the former lemma ensures that all of the components are O(|\F|^{-1/4})-regular. This lemma, which may be of independent interest, relies on some basic facts about the \'etale fundamental group of an algebraic variety
Expanding polynomials on sets with few products
In this note, we prove that if is a finite set of real numbers such that
, then for every polynomial we have that
, unless is of the form
for some monomial and some univariate polynomial
Expanding Polynomials on Sets with Few Products
In this note, we prove that if A is a finite set of real numbers such that |AA|=K|A|, then for every polynomial f∈R[x,y] we have that |f(A,A)|=Ω_(K,degf)(|A|²), unless f is of the form f(x,y)=g(M(x,y)) for some monomial M and some univariate polynomial g. This is sharp up to the dependence on K and the degree of f
Strongly minimal pseudofinite structures
We observe that the nonstandard finite cardinality of a definable set in a
strongly minimal pseudofinite structure D is a polynomial over the integers in
the nonstandard finite cardinality of D. We conclude that D is unimodular,
hence also locally modular. We also deduce a regularity lemma for graphs
definable in strongly minimal pseudofinite structures. The paper is elementary,
and the only surprising thing about it is that the results were not explicitly
noted before. In the new version we add a comment on relations to work of
Macpherson and Steinhorn, as well as on the limited nature of the examplesComment: 11 page