On a theorem of Sárközy for difference sets and shifted primes

@article{Wang2020OnAT,
  title={On a theorem of S{\'a}rk{\"o}zy for difference sets and shifted primes},
  author={Ruoyi Wang},
  journal={Journal of Number Theory},
  year={2020}
}
  • R. Wang
  • Published 8 June 2019
  • Mathematics
  • Journal of Number Theory
Multivariate Polynomial Values in Difference Sets
TLDR
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References

SHOWING 1-6 OF 6 REFERENCES
Difference sets and the primes
Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.
Difference sets and shifted primes
We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)−c(log log log log log n)) for
Analytic Number Theory
Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large
On sets of natural numbers whose difference set contains no squares
We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous