Naslov
A strong-type Furstenberg-Sárközy theorem for sets of positive measure

Autori
Durcik, Polona ; Kovač, Vjekoslav ; Stipčić, Mario

Izvornik
Journal of geometric analysis (1050-6926) 33 (2023), 8; 255, 16

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
non-linear patterns ; maximal functions ; density theorem ; Euclidean space

Sažetak
For every β∈(0, ∞), β≠1 we prove that a positive measure subset A of the unit square contains a point (x_0, y_0) such that A nontrivially intersects curves y−y_0=a(x−x_0)^β for a whole interval I⊆(0, ∞) of parameters a∈I. A classical Nikodym set counterexample prevents one to take β=1, which is the case of straight lines. Moreover, for a planar set A of positive density we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA