An extension of the Erdős‐Tetali theorem

Abstract

Given a sequence 𝒜=false{a0<a1<a2⋯false}⊆double-struckN, let r𝒜,h(n) denote the number of ways n can be written as the sum of h elements of 𝒜. Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O‐regularly varying and of positive increase) satisfying x1/hlog(x)1/h≪f(x)≪x1/(h−1)log(x)εfor someε>0, then there must exist 𝒜⊆double-struckN with false|𝒜∩false[0,xfalse]false|=normalΘfalse(ffalse(xfalse)false) for which r𝒜,h + ℓ(n) = Θ(f(n)h + ℓ/n) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to x1false/2normallogfalse(xfalse)1false/2≪ffalse(xfalse)≪x. The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.

An elementary heuristic for Hardy–Littlewood extended Goldbach’s conjecture

“…Section 38). In Section III of Halberstam & Roth[14] one finds this construction for the specific case we are working with.4 See Theorem 3.1 in[23] for a short proof of this version of the strong law.…”