Abstract
Let \(\mathcal{P}\) denote the set of primes. For a fixed dimension \(d\), Cook– Magyar–Titichetrakun, Tao–Ziegler and Fox–Zhao independently proved that any subset of positive relative density of \(\mathcal{P}^d\) contains an arbitrary linear configuration. In this paper, we prove that there exists such configuration with the step being a shifted prime (prime minus \(1\) or plus \(1\)).
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Acknowledgement
The authors would like to thank Bryna Kra for useful feedback and the referee for helpful suggestions. The first author also thanks the hospitality of the University of Mississippi where part of this work was carried out.
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The second author is supported by National Science Foundation Grant DMS-1702296.
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Le, A.N., Lê, T.H. Multidimensional configurations in the primes with shifted prime steps. Acta Math. Hungar. 164, 1–14 (2021). https://doi.org/10.1007/s10474-021-01143-9
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DOI: https://doi.org/10.1007/s10474-021-01143-9