The sum-product problem for integers with few prime factors
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909668742266880 |
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| author | Hanson, Brandon Rudnev, Misha Shkredov, Ilya Zhelezov, Dmitrii |
| author_facet | Hanson, Brandon Rudnev, Misha Shkredov, Ilya Zhelezov, Dmitrii |
| contents | It was asked by E. Szemerédi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $ω(a)\leq k$. In this paper, answer Szemerédi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-ε}$ for some $ε>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04038 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The sum-product problem for integers with few prime factors Hanson, Brandon Rudnev, Misha Shkredov, Ilya Zhelezov, Dmitrii Number Theory Combinatorics It was asked by E. Szemerédi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $ω(a)\leq k$. In this paper, answer Szemerédi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-ε}$ for some $ε>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. |
| title | The sum-product problem for integers with few prime factors |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2305.04038 |