Arithmetic Progressions in Dense Sets Article Swipe
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· 2026
· Open Access
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· DOI: https://doi.org/10.5281/zenodo.18119725
· OA: W7117977844
The study of arithmetic progressions within dense subsets of the integers constitutes a cornerstoneof modern additive combinatorics, bridging elementary number theory, harmonic analysis, ergodictheory, and graph theory. This paper provides a rigorous, comprehensive examination of thequantitative bounds associated with Szemer´edi’s Theorem, which asserts that any subset of theintegers with positive upper density contains arbitrarily long arithmetic progressions. We focusspecifically on the evolution of upper bounds for the size of subsets of {1, . . . , N } lacking k-termarithmetic progressions, denoted as rk(N ).Beginning with the foundational work of Roth regarding 3-term progressions, we trace themethodological advancements from classical Fourier analysis on ZN to the higher-order Fourieranalysis pioneered by Gowers. The abstract nature of the problem is dissected through the lensof the Gowers uniformity norms U k, which provide a measure of pseudorandomness necessary tocontrol the counting of linear configurations. We analyze the dichotomy between structure andrandomness, a central theme in the field, and how it is exploited via the density increment strategy.Furthermore, this manuscript investigates the recent breakthroughs in breaking the logarithmicbarrier for Roth’s Theorem and the subsequent improvements for general k. We synthesize thecontributions of the polynomial method, the inverse theorems for Gowers norms, and the trans-ference principles that allowed for the extension of these results to sparse sets, such as the primes(the Green-Tao Theorem).
