On solution-free sets for simultaneous diagonal polynomials

2010/10/08 by M. Smith, Smith, Matthew L.
Computer Science · Mathematics · #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1010.1575

openalex publication_date 2010/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle method that if a subset A of the natural numbers restricted to the interval [1,N] satisfies Gowers' definition of uniformity of degree k, then it furnishes roughly the expected number of simultaneous solutions to the given equations. If A furnishes no non-trivial solutions to the given system, then we show that the number of elements of A in [1,N] grows no faster than a constant multiple of N/(log log N)^-c as N grows to infinity, where c>0 is a constant dependent only on k. In particular, we show that the density of A in [1,N] tends to 0 as N tends to infinity.

Citations

• On Some Sequences of Integers
• Roth’s theorem on progressions revisited
• On Certain Sets of Integers

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