CONJECTURES OF DIOPHANTINE EQUATIONS ON PIATETSKI-SHAPIRO SEQUENCES (Analytic Number Theory and Related Topics)

Abstract

Let α > 1 be a non-integral real number. Let PS(α) be the set of positive integers of the form ⌊[α]𝓃 ⌋ for some 𝓃 ∈ ℕ. In this article, we discuss the equation 𝓍 + 𝓎 = 𝓏, where (𝓍, 𝓎, 𝓏) ∈ PS(α)³. The author conjectures that for almost all or all 2 < α < 3 the equation 𝓍 + 𝓎 = 𝓏 has infinitely many solutions (𝓍, 𝓎, 𝓏) ∈ PS(α)³. In this article, we aim to present heuristic and numerical evidence of the conjecture.