EFFECTIVE SCHNIRELMANN'S METHOD FOR O-REGULAR SEQUENCES CHRISTIAN TAFULA´ Abstract. In this paper we introduce the notion of pre-basis, which is a sequence such that some of its h-fold sumsets has positive lower asymptotic density, the least such h being its pre-order. Schnirelmann's classical theory of sumsets is reinterpreted as characterizing additive bases in terms of pre-bases, and estimates for the order in terms of the pre-order are derived from the deep theorems of Mann and Kneser. Under certain regularity assumptions on the representation functions, we then derive estimates to the pre-order. This is achieved by studying sequences A = fa0 < a1 < a2 < : : :g ⊆ N for which A(2x) = O(A(x)) and a2n = O(an), and ends up providing a small shortcut to the proofs of the Schnirelmann-Goldbach theorem and Linnik's elementary solution of Waring's problem. 1. Introduction Given A = fa0 < a1 < a2 < : : :g ⊆ N an infinite sequence and h ≥ 2 an Ph integer, the h-fold sumset hA is the set f i=1 ki : k1; : : : ; kh 2 A g. We say that A is an additive basis (resp. additive asymptotic basis) when there is h ≥ 2 such that N n hA is empty (resp. finite), and the least such h is called the order (resp. asymptotic order) of A , denoted by O(A ) (resp. O(A )). The representation functions rA ;h(n) and sA ;h(x) count the number of solutions of k1 + k2 + ::: + kh = n and k1 + k2 + ::: + kh ≤ x resp., where h ≥ 1 is a fixed integer and ki 2 A , considering permutations in the sense of the formal series: h ! P ah X X z X (1.1) za = r (n)zn; a2A = s (n)zn: A ;h 1 − z A ;h a2A n≥0 n≥0 Certain regularity properties on representation functions are essential to the elementary arguments showing that primes (Schnirelmann-Goldbach theorem, cf. Chapter 7 of Nathanson [4]) and sequences generated by poly- nomials (Generalized Waring's problem, cf. Chapters 11 & 12 of Nathanson [6]) are additive bases. In this paper we investigate some of these regularity properties in a general setting. Denoting by A(x) the quantity jA \ [0; x]j, we study three types of sequences: the O-regular sequences (OR), which satisfy A(2x) = O(A(x)); the positively increasing sequences (PI), which satisfy a2n = O(an); and the 2010 Mathematics Subject Classification. Primary 11B13, 11B34. Key words and phrases. additive bases, representation functions, O-regular variation, Schnirelmann's theorem, Schnirelmann's method. 1 2 CHRISTIAN TAFULA´ O-regular plus sequences (OR+), which are sequences being both OR and PI. The first part of this work is dedicated to motivate the study of these sequences by the asymptotic behavior of their representation functions. In the second part we introduce the notion of additive pre-bases. Given a sequence A ⊆ N, consider the following densities: Schnirelmann density: σ(A ) := inf (A(n) − A(0))=n; n≥1 Asymptotic density: d(A ) := lim A(x)=x; x!+1 Power density: w(A ) := lim log(A(x))= log(x); x!+1 the last two having lower (lim inf) and upper (lim sup) forms, signalized by a lower or upper bar, resp. (e.g. w, d). We define the Schnirelmann pre-order and pre-order of A as, respectively, Oe(A ) := minfh ≥ 1 : σ(hA ) > 0g; Oe(A ) := minfh ≥ 1 : d(hA ) > 0g: Thus pre-bases are sequences A with finite pre-order. The motivation for this concept comes from Schnirelmann's characterization of additive bases (see Theorem 3.1), which basically states: • B ⊆ N is a basis () Oe(B) < +1 and f0; 1g ⊆ B; • A ⊆ N is an asymptotic basis () Oe(A ) < +1 and A is not contained in an arithmetic progression with common difference r ≥ 2. Our main result is a combination of effective bounds for the order of a sequence in terms of its pre-order and a regularity condition on representation functions that ensures a sequence is a pre-basis. This last ingredient relates to what is loosely referred to in the literature as Schnirelmann's method. In order to state it, say a pre-basis A is stable when every A 0 ⊆ A with A0(x) = Θ(A(x)) is also a pre-basis. Moreover, say it is uniformly stable when there is h ≥ 1 such that Oe(A 0) ≤ h for every such A 0 ⊆ A . Main Theorem (Effective Schnirelmann's method). Let A ⊆ N be an OR+ sequence. If for some h ≥ 1 we have A(n)h X r (n) ξ(n); where ξ(n)2 = O(x); A ;h n n≤x then A is a uniformly stable pre-basis, with Oe(A 0) ≤ h for every subsequence A 0 ⊆ A with A0(x) = Θ(A(x)). Furthermore: • If f0; 1g ⊆ A , then the order of A is at most dσ(hA )−1eh; • If A is not contained in an arithmetic progression, then the asymp- totic order of is at most dd(h )−1 + 1 ( 1−d(hA ) )eh + bd(h )ch. A A h d(hA ) A This statement is the combination of Corollary 3.4 with Theorem 3.1, and it constitutes a small shortcut to the proof of the Schnirelmann-Goldbach theorem and the elementary solution of the generalized Waring's problem, covering their relatively easy parts. EFFECTIVE SCHNIRELMANN'S METHOD FOR O-REGULAR SEQUENCES 3 Notation. A real function f is asymptotically defined when f is well-defined in [α; +1) for some α ≥ 0. Whenever we write an \α" not specified at the context, this is the meaning that will be implicitly implied. Our use of asymptotic notation (Θ; ; O; ; o; ∼) is standard, with the addition of \≺", which has the same meaning as small-o (i.e. f ≺ g () f = o(g))1. 2. O-regularity in sequences We start with some generalities. Given a sequence A ⊆ N, let n 2 N and h ≥ 2 an integer. Then, from the formal series (1.1) one can immediately deduce the following recursive formulas: X rA ;h(n) = rA ;h−`(k)rA ;`(n − k) k≤n (2.1) X X sA ;h(n) = rA ;h−`(k)sA ;`(n − k) = sA ;h−`(k)rA ;`(n − k) k≤n k≤n For purposes of induction the case ` = 1 is usually enough, and we denote 1 rA ;1(n) simply by A (n). The following proposition motivates our study of O-regularity in sequences. Proposition 2.1. For every sequence A ⊆ N, sA ;h+1(x) sA ;h(x); 8h ≥ 1: Proof. Choose some 0 < " < 1 and define: 1−" L1(x) := A(x) ; 1−" Lh+1(x) := A(Lh(x)) ; 8h ≥ 1: We will use induction to show that, for all h ≥ 1, sA ;h+1(x) (2.2) sA ;h(x) : A(Lh(x)) As Lh+1(x) ≺ Lh(x) and Lh(x) 1 for all h, it will imply the statement of our proposition. Let us start with h = 1. Given that X 1 sA ;1(x) − sA ;1(x − L1(x)) = A (n) ≺ sA ;1(x); x−L1(x)<n≤x it follows that sA ;1(x) ∼ sA ;1(x − L1(x)). Hence, by the recursive formulas in (2.1), X 1 sA ;2(n) = sA ;1(n − k) A (k) k≤n X 1 sA ;1(n − k) A (k) k≤L1(n) sA ;1(n − L1(n)) · A(L1(n)) sA ;1(n) · A(L1(n)); thus (2.2) holds for h = 1. 1This is one of G. H. Hardy's asymptotic symbols. Despite being a little old-fashioned, we think it provides a nice counterpart for small-o, in the same way \" does for Big-O. 4 CHRISTIAN TAFULA´ For the induction step, note that for every h ≥ 1 we have X 1 (2.3) rA ;h+1(n) = rA ;h(k) A (n − k) ≤ sA ;h(n) k≤n Taking h > 1 and assuming (2.2) valid for h − 1, we have, by (2.3), X sA ;h(x) − sA ;h(x − Lh(x)) = rA ;h(n) x−Lh(x)<n≤x X ≤ sA ;h−1(n) x−Lh(x)<n≤x Lh(x)sA ;h−1(x) sA ;h(x) Lh(x) ≺ sA ;h(x); A(Lh−1(x)) then it follows that sA ;h(x) ∼ sA ;h(x − Lh(x)). Hence: X 1 sA ;h+1(n) = sA ;h(n − k) A (k) k≤n X 1 sA ;h(n − k) A (k) k≤Lh(n) sA ;h(n − Lh(n)) · A(Lh(n)) sA ;h(n) · A(Lh(n)); thus (2.2) holds for all h. This is a quite general estimate, and with the right restraints on the growth of A(x) we can actually say substantially more. 2.1. OR sequences. Let us introduce some bits of regular variation theory. An extensive treatment on this topic can be found in Bingham, Goldie & Teugels [1]. We will only need the theory from Chapters 2. Take f an asymptotically defined positive real function. We say f is • Slowly varying if f(λx) ∼ f(x) for all λ > 0; • Regularly varying if f(λx) ∼ λρf(x), for all λ > 0 and some ρ 2 R; • O-regularly varying if f(λx) f(x) for all λ > 0. The generality of these definitions lies on Karamata's characterization theorem2, which states that if f(λx) ∼ g(λ)f(x) with g(λ) 2 (0; +1) for all λ > 0 and f is measurable, then f is regularly varying, i.e. there is ρ 2 R such that g(λ) ≡ λρ.