Year: 2010
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Smith, Matthew L.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1642453141