New Results for the Growth of Sets of Real Numbers

Abstract

We use the theory of cross ratios to construct a real-valued function f of only three variables with the property that for any finite set A of reals, the set $$f(A)=\{f(a,b,c):a,b,c \in A\}$$f(A)={f(a,b,c):a,b,c?A} has cardinality at least $$C|A|^2/\log |A|$$C|A|2/log|A|, for an absolute constant C. Previously-known functions with this property have all been of four variables. We also improve on the state of the art for functions of four variables by constructing a function g for which g(A) has cardinality at least $$C|A|^2$$C|A|2; the previously best-achieved bound was $$C|A|^2/\log |A|$$C|A|2/log|A|. Finally, we give an example of a five-variable function h for which h(A) has cardinality at least $$C|A|^4/\log |A|$$C|A|4/log|A|. Proving these results depends only on the Szemeredi---Trotter incidence theorem and an analoguous result for planes due to Edelsbrunner, Guibas and Sharir, each applied in the Erlangen-type framework of Elekes and Sharir. In particular the proofs do not employ the Guth---Katz polynomial partitioning technique or the theory of ruled surfaces. Although the growth exponents for f, g and h are stronger than those for previously-considered functions, it is not clear that they are necessarily sharp. So we pose a question as to whether the bounds on the cardinalities of f(A), g(A) and h(A) can be further strengthened.