An extension of the Erdős‐Tetali theorem. Issue 1 (19th November 2018) - Táfula, Christian
Abstract : Given a sequence 𝒜 = a 0 < a 1 < a 2 ⋯ ⊆ N, let r 𝒜, 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 x 1 / h log ( x ) 1 / h ≪ f ( x ) ≪ x 1 / ( h − 1 ) log ( x ) ε for some ε > 0, then there must exist 𝒜 ⊆ N with | 𝒜 ∩ [ 0, x ] | = Θ ( f ( x ) ) for which r 𝒜, h + ℓ ( n ) = Θ( f ( n ) h + ℓ / n ) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to x 1 / 2 log ( x ) 1 / 2 ≪ f ( x ) ≪ 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.