Sums and products in sets of positive density
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915782797033472 |
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| author | Richter, Florian K. |
| author_facet | Richter, Florian K. |
| contents | We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish two main results. First, we show that if $Q(1) = 0$, then any set of natural numbers with positive upper logarithmic density contains a pair of the form $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N} \setminus \{1\}$. Second, we prove that if $Q(0) = 0$, then any set of natural numbers with positive density relative to a new multiplicative notion of density, which arises naturally in the context of such problems, contains $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums and products in sets of positive density Richter, Florian K. Combinatorics Dynamical Systems Number Theory 05D10, 37A44, 11B30 We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish two main results. First, we show that if $Q(1) = 0$, then any set of natural numbers with positive upper logarithmic density contains a pair of the form $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N} \setminus \{1\}$. Second, we prove that if $Q(0) = 0$, then any set of natural numbers with positive density relative to a new multiplicative notion of density, which arises naturally in the context of such problems, contains $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N}$. |
| title | Sums and products in sets of positive density |
| topic | Combinatorics Dynamical Systems Number Theory 05D10, 37A44, 11B30 |
| url | https://arxiv.org/abs/2507.00515 |