Abstract
Let G be a multiplicative subgroup of the prime field F p of size |G| > p1−κ and r an arbitrarily fixed positive integer. Assuming κ = κ(r) > 0 and p large enough, it is shown that any proportional subset A ⊂ G contains non-trivial arithmetic progressions of length r. The main ingredient is the Szemerédi–Green–Tao theorem.
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Research partially financed by the NSF Grants DMS 1600154.
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Chang, MC. Arithmetic progressions in multiplicative groups of finite fields. Isr. J. Math. 222, 631–643 (2017). https://doi.org/10.1007/s11856-017-1602-z
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DOI: https://doi.org/10.1007/s11856-017-1602-z