Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Narrow progressions in the primes (1409.1327v2)

Published 4 Sep 2014 in math.NT

Abstract: In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = {1,\dots,N}$ of relative density at least $\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$ in $A$ with $0 < |r| \leq N{o(1)}$, if $N$ is sufficiently large depending on $k,P_1,\dots,P_k$ and $\delta$. In this paper we shorten the size of this progression to $0 < |r| \leq \logL N$, where $L$ depends on $k,P_1,\dots,P_k$ and $\delta$. In the linear case $P_i = (i-1)\mathbf{m}$, we can take $L$ independent of $\delta$. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions.

Summary

We haven't generated a summary for this paper yet.