Leírás
Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ``arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\in\mathbb{Z}[x]$ and obtain a quasipolynomial upper bound for the largest subset of $\{1,2,\dots,X\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\in \Z$. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences. In contrast to the square case, extending the method to general intersective polynomials requires performing a density increment iteration in which the underlying polynomial changes at each step; a key contribution of this paper is to show that the arithmetic level-$d$ inequality remains effective uniformly across all auxiliary
polynomials arising in the iteration. We also develop smoothly weighted versions of the exponential sum estimates of Rice.
This is joint work with eight other authors, all of whom are either PhD students or postdocs at the University of Georgia:
Carlo Francisco E. Adajar
Rishika Agrawal
Mukul Rai Choudhuri
Chian Yeong Chuah
Steve Fan
Swaroop Hegde
Krishnamohan Nandakumar
Nagendar Reddy Ponagandla
április 7., 14:15, Rényi Intézet Nagyterem (Main Lecture Hall)
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