Sárközy's theorem for shifted primes with restricted digits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915555595780096 |
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| author | Burgin, Alex |
| author_facet | Burgin, Alex |
| contents | For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has positive upper Banach density, then there exists a prime $p\in \mathcal{P}$ and two elements $a_1,a_2\in A$ such that $a_2=a_1+p-1$. The key ingredients are the Furstenberg correspondence principle and a discretized Hardy-Littlewood circle method used by Maynard. As a byproduct of our work, we prove a Dirichlet-type theorem for the distribution of $\mathcal{P}$ in residue classes, and a Vinogradov-type theorem for the decay of associated exponential sums. These estimates arise from the unique structure of associated Fourier transforms, which take the form of Riesz products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sárközy's theorem for shifted primes with restricted digits Burgin, Alex Number Theory For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has positive upper Banach density, then there exists a prime $p\in \mathcal{P}$ and two elements $a_1,a_2\in A$ such that $a_2=a_1+p-1$. The key ingredients are the Furstenberg correspondence principle and a discretized Hardy-Littlewood circle method used by Maynard. As a byproduct of our work, we prove a Dirichlet-type theorem for the distribution of $\mathcal{P}$ in residue classes, and a Vinogradov-type theorem for the decay of associated exponential sums. These estimates arise from the unique structure of associated Fourier transforms, which take the form of Riesz products. |
| title | Sárközy's theorem for shifted primes with restricted digits |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.13076 |