Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866914569904979968 |
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| author | Adajar, Carlo Francisco E. Agrawal, Rishika Choudhuri, Mukul Rai Chuah, Chian Yeong Fan, Steve Hegde, Swaroop Lott, Andrew Nandakumar, Krishnamohan Ponagandla, Nagendar Reddy |
| author_facet | Adajar, Carlo Francisco E. Agrawal, Rishika Choudhuri, Mukul Rai Chuah, Chian Yeong Fan, Steve Hegde, Swaroop Lott, Andrew Nandakumar, Krishnamohan Ponagandla, Nagendar Reddy |
| contents | Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--Sárközy 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 an analogous 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 $n\in \mathbb{Z}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality Adajar, Carlo Francisco E. Agrawal, Rishika Choudhuri, Mukul Rai Chuah, Chian Yeong Fan, Steve Hegde, Swaroop Lott, Andrew Nandakumar, Krishnamohan Ponagandla, Nagendar Reddy Number Theory Combinatorics 11P55 Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--Sárközy 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 an analogous 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 $n\in \mathbb{Z}$. 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. |
| title | Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality |
| topic | Number Theory Combinatorics 11P55 |
| url | https://arxiv.org/abs/2605.16216 |