Improved Bounds for Szemerédi's Theorem (2402.17995v2)

Abstract: Let $r_k(N)$ denote the size of the largest subset of $[N] = {1,\ldots,N}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that [r_k(N)\ll N\exp(-(\log\log N)^{c_k}).] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemer\'{e}di as reformulated by Green and Tao.

• The paper improves the upper bounds for rₖ(N) by proving rₖ(N) ≪ N exp(–(log log N)^(cₖ)), significantly refining previous polynomial decay limits.
• It employs a novel global inverse theorem approach that utilizes nilsequence structures and iteratively refines bracket expressions to manage complexity.
• The study integrates advanced techniques in higher order Fourier analysis and density increment strategies, paving the way for broader applications in arithmetic combinatorics.

Improved Bounds for Szemerédi's Theorem

The paper investigates the mathematical bounds associated with Szemerédi's Theorem, specifically focusing on enhancing the established limits concerning the size of subsets of the natural numbers that avoid containing $k$-term arithmetic progressions, denoted as $r_k(N)$. For a given $k \ge 5$, the authors demonstrate that $r_k(N) \ll N \exp(-(\log\log N)^{c_k})$ for some constant $c_k>0$. This is a significant improvement upon previous results by leveraging advancements in the quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm and the density increment strategies originating from Heath-Brown and Szemerédi, further refined by Green and Tao.

Main Contributions

1. Upper Bound Improvements: The authors refine the upper bounds for $r_k(N)$, advancing on the original contributions from Szemerédi and subsequent improvements by Gowers which resulted in polynomial decay rates with respect to logarithmic double iterated behaviors. This paper pushes these boundaries further by utilizing advanced methods in higher order Fourier analysis and density increment strategies.
2. Tool Development: Key to this improvement is the adaptation and application of global inverse theorems, rather than local theorems, which create more robust correlations and make use of the comprehensive structure of nilsequences instead of polynomial sequences alone.
3. Schmidt-type Problem Resolution: A unique challenge in the approach is the decomposition of polynomial sequences on nilmanifolds, requiring sophisticated solutions to Schmidt-type decomposition problems. The authors present a novel iterative technique that addresses this through an inside-out approach, effectively managing nested bracket expressions.
4. Iterative Schmidt Refinement: The paper introduces a method of iteratively “reducing” bracket expressions to control sequences and avoid complex nested integer operations. This refinement strategy allows for an orderly reduction of nilsequences, maintaining the sequences' bounding within controlled deviations and thus supporting robust progression towards improved density increments.

Theoretical and Practical Implications

The theoretical implications of this work lie in its potential to influence and extend research in arithmetic combinatorics, specifically through the prism of understanding patterns within large sets of integers. The convergence of density increment strategies with inverse theorems represents a robust technique for tackling similar bounds in mathematical settings involving combinatorial constraints.

Practically, although the immediate applications of such pure mathematical advancements might not be evident, they frequently influence fields requiring optimized genetic algorithms, pseudo-random generation methodologies, and complex networks theory, where understanding and controlling progression-related behaviors is crucial.

Future Directions

Considering the depth of advancements introduced by this research, future studies could continue to investigate:

• Extension to More General Structures: Applying these concepts to broader mathematical structures, potentially expanding beyond integers to other algebraic structures or settings, thereby testing the versatility of the methods developed in this paper.
• Computational Applications: Translating these abstract bounds into computational algorithms that can explore randomness and distribution in data sets more efficiently, as guided by these combinatorial insights.
• Cross-disciplinary Extensions: Examining intersections with other domains such as signal processing and data compression where approximation of subsets and increase in density play critical roles.

Overall, the paper not only strengthens Szemerédi's Theorem in its traditional form but also sets significant precedence for future mathematical inquiries spurred by the synthesis of combinatorial and analytic strategies in number theory and beyond.