[Paper Review] Polynomial largeness of sumsets and totally ergodic sets

Alexander Fish|ArXiv.org|Nov 20, 2007

Limits and Structures in Graph Theory4 references3 citations

TL;DR

This paper establishes that sumsets of totally ergodic (TE) sets in the natural numbers intersect the values of any non-constant polynomial with integer coefficients, provided the other set has positive upper density. It further proves that for weakly mixing (WM) sets—a subclass of TE sets—the intersection has lower density 1, generalizing to multiple polynomials and sets via ergodic-theoretic techniques and van der Corput-type estimates.

ABSTRACT

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE set, for any (p(n) \in \Z[n]: °{p(n)} > 0, p(n) o_{n o \infty} \infty ) and any subset (B \subset \N ) of positive upper density we have (R_p = A+B \cap \{p(n) | n \in \N \} eq \emptyset). For (A ) a WM set (subclass of TE sets) we prove that (R_p ) has lower density 1. In addition we obtain a generalization of the latter result to the case of several polynomials and several WM sets.

Motivation & Objective

• To determine whether aperiodic sets (TE sets) ensure sumset intersections with polynomial values for any positive-density set.
• To investigate whether weakly mixing (WM) sets, a subclass of TE sets, yield stronger intersection properties, such as lower density 1.
• To generalize the result to multiple polynomials and multiple WM sets using dynamical systems and averaging techniques.
• To establish a connection between topological dynamics (ergodicity, weak mixing) and additive combinatorics in the context of polynomial images in sumsets.

Proposed method

• Uses the correspondence between characteristic sequences of subsets of ℕ and dynamical systems on the Cantor space {0,1}^ℕ via the shift map.
• Applies ergodic-theoretic definitions: generic points, total ergodicity, and weak mixing to characterize TE and WM sets.
• Employs van der Corput-type estimates (Lemma 5.1) to control uniform distribution of sequences in Hilbert space.
• Uses weighted average convergence in L² for weakly mixing systems (Lemma 5.2) to handle non-uniform weights in averaging.
• Applies induction and contradiction arguments on nested sums involving characteristic functions of sets and polynomial shifts.
• Leverages the fact that for WM systems, the inner product of shifted functions converges to zero in L², enabling density estimates.

Experimental results

Research questions

• RQ1 Does every TE set A intersect A + B with the image of any non-constant integer polynomial p(n) when B has positive upper density?
• RQ2 For WM sets A, does the sumset A + B intersect the image of p(n) with lower density 1?
• RQ3 Can the result be extended to multiple polynomials and multiple WM sets simultaneously?
• RQ4 What dynamical properties (e.g., total ergodicity, weak mixing) ensure that sumsets avoid polynomial images?

Key findings

• Any TE set A intersects A + B with the image of any non-constant polynomial p(n) ∈ ℤ[n] with positive leading coefficient, provided B has positive upper density.
• For WM sets A, the intersection (A + B) ∩ {p(n) | n ∈ ℕ} has lower density 1 when B has positive density.
• The result generalizes to k+1 polynomials of the same degree and k+1 WM sets, where the solution set to the additive system has lower density 1.
• The proof relies on contradiction via boundedness from below of a certain sum involving characteristic functions and polynomial shifts, contradicting a smallness estimate from van der Corput's lemma.
• The key technical step uses that in weakly mixing systems, weighted averages of shifts converge to zero in L², enabling density control.
• The lower density 1 result holds even when the density of B exists and is positive, and the polynomial image is not required to be dense in any arithmetic progression.