FRACTIONAL PARTS OF POLYNOMIALS OVER THE PRIMES. II. Issue 3 (26th June 2018) - Baker, Roger
Abstract : Let | ⋯ | denote distance from the integers. Let α, β, γ be real numbers with α irrational. We show that the inequality ‖ α p 2 + β p + γ ‖ < p − 3 / 17 + ε has infinitely many solutions in primes p, sharpening a result due to Harman [On the distribution of α p modulo one II. Proc. Lond. Math. Soc. (3) 72 (1996), 241–260] in the case β = 0 and Baker [Fractional parts of polynomials over the primes. Mathematika 63 (2017), 715–733] in the general case.