Product sets cannot contain long arithmetic progressions
Dmitry Zhelezov

TL;DR
This paper establishes upper bounds on the length of arithmetic progressions within product sets of natural numbers and complex numbers, revealing limitations on their structure and providing new insights into their combinatorial properties.
Contribution
It proves new upper bounds on the longest arithmetic progression in product sets of natural and complex numbers, advancing understanding of their combinatorial structure.
Findings
Longest arithmetic progression in natural number product sets is at most O(n log^2 n / log log n)
Existence of product sets with arithmetic progressions of length Ω(n log n)
Upper bound for complex number product sets is O(n^{3/2})
Abstract
Let be a set of natural numbers of size . We prove that the length of the longest arithmetic progression contained in the product set cannot be greater than and present an example of a product set containing an arithmetic progression of length . For sets of complex numbers we obtain the upper bound .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · Mathematical Dynamics and Fractals
