On thin sum-product bases

Abstract

Additive bases, and less importantly multiplicative bases, have been ex- tensively studied for several centuries. More recently, expanding polynomi- als (of course, with more than one variable) have been considered with a view to studying the expansion of nite sets under these polynomials. If f 2Z[x1;x2; : : : ;xd] and A is contained in a given subset R of a commutative ring, then let f(A;A;: : : ;A) (with k arguments) denote the set of all terms f(a1;a2; : : : ;ak) where the ai's belong to A. The polynomial f is called an expander if there exists >0 such that jf(A;: : : ;A)j>jAj1+ for any nite set A, where jBj denotes the cardinality of a nite set B. If R is nite, as for instance, if R=Fq or f1; : : : ;Ng, we need to restrict the above de nition by assuming that jRj"<jAj<jRj1􀀀", for some ">0. A more restrictive no- tion is of a covering polynomial which arises from the following question: is there a non trivial minimal size such that if A attains it, then f(A;A;: : : ;A) entirely covers R?