Approximate Polynomial Structure in Additively Large Sets

In [1], the authors introduced a measure space, obtained by taking a quotient of a Loeb measure space, that has the property that multiplication is measure-preserving and for which standard sets of positive logarithmic density have positive measure. The log Banach density of a standard set (see Section 2 below for the definition) was also introduced, and this measure space framework was used, in conjunction with Furstenberg’s Recurrence Theorem, to obtain a standard result about the existence of approximate geometric progressions in sets of positive log Banach density. In this paper, we improve the bounds of approximation of this result by using Szemerédi’s Theorem together with a “logarithmic change of coordinates.” More specifically, in Proposition 3.1, we show that if A is a standard subset of the natural numbers, then the Banach density of {⌈log2(x)⌉ : x ∈ A} is greater than or equal to the log Banach density of A. This allows us to use Szemerédi’s Theorem to show that every set of positive Banach log density contains a set which is “within a factor of 2” of being a geometric sequence; Theorem 3.3 provides a precise version of this statement. We also explore a family of densities on the natural numbers, the (upper) r-Banach densities for 0 < r ≤ 1, which have the property that positive 1/m-Banach density implies the existence of approximate mth powers of arithmetic progressions, in a sense made precise in Theorem 3.7. (This family of densities was introduced in [1], although BDm(A) in that paper corresponds to BD1/m(A) here.) In Section 2 we establish some properties of the log Banach density and the r-Banach densities, most notably that the log Banach density of a set A is always less than or