## Abstract

We prove a transference type result for pseudo-random subsets of

ZN that is analogous to the well-known Furstenberg-S ¨ ark ´ ozy theorem. More pre- ¨

cisely, let k ≥ 2 be an integer and let β and γ be real numbers satisfying

γ + (γ − β)/(2k+1 − 3) > 1.

Let ⊆ ZN be a set with size at least Nγ and linear bias at most Nβ. Then, every

A ⊆ with relative density |A|/|| ≥ (log log N)

− 1

2 log log log log log N contains

a pair of the form {x, x + dk } for some nonzero integer d.

For instance, for squares, i.e., k = 2, and assuming the best possible pseudorandomness β = γ /2 our result applies as soon as γ > 10/11.

Our approach uses techniques of Green as seen in [6] relying on a Fourier

restriction type result also due to Green

ZN that is analogous to the well-known Furstenberg-S ¨ ark ´ ozy theorem. More pre- ¨

cisely, let k ≥ 2 be an integer and let β and γ be real numbers satisfying

γ + (γ − β)/(2k+1 − 3) > 1.

Let ⊆ ZN be a set with size at least Nγ and linear bias at most Nβ. Then, every

A ⊆ with relative density |A|/|| ≥ (log log N)

− 1

2 log log log log log N contains

a pair of the form {x, x + dk } for some nonzero integer d.

For instance, for squares, i.e., k = 2, and assuming the best possible pseudorandomness β = γ /2 our result applies as soon as γ > 10/11.

Our approach uses techniques of Green as seen in [6] relying on a Fourier

restriction type result also due to Green

Original language | English |
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Title of host publication | The Seventh European Conference on Combinatorics, Graph Theory and Applications |

Pages | 421-424 |

Number of pages | 4 |

Volume | 16 |

State | Published - 2013 |