Sarkozy's Theorem for P-Intersective Polynomials

Details Sarkozy's Theorem for P-Intersective Polynomials Sarkozy's Theorem for P-Intersective Polynomials

2011-11-28

arxiv.org/abs/1111.6559v3

Mathematics

Classical Analysis and ODEs

15 pages, argument modified to give order of growth estimate in Theorem 4, Theorems 2 and 3 added

Scientific paper

We define a necessary and sufficient condition on a polynomial h\in \Z[x] to guarantee that every set of natural numbers of positive upper density and every set of primes of positive relative upper density contains a nonzero difference of the form h(p) for some prime p. Moreover, we establish a quantitative estimate on the size of the largest subset of ${1,2,...,N}$ which lacks the desired arithmetic structure, showing that if deg(h)=k, then the density of such a set is at most a constant times (\log N)^{-c} for any c<1/(2k-2). We also discuss how an improved version of this result for k=2 and a relative version in the primes can be obtained with some additional known methods.

Affiliated with

Also associated with

No associations

Rating

Sarkozy's Theorem for P-Intersective Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Sarkozy's Theorem for P-Intersective Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sarkozy's Theorem for P-Intersective Polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-68457