[Sh] Pablo Shmerkin, On the exceptional set for abso- new density for a set of integers that is exactly the lute continuity of Bernoulli convolutions, Geom. right density for the investigation of additive bases. Funct. Anal. 24 (2014), 946–958. (For a survey of the classical bases in additive [So] Boris Solomyak, On the random series n (an ± number theory, see Nathanson [28].) Erd˝osproblem), Ann. of Math. (2), 142 P(3) (1955), 611–625. [W] Hermann Weyl, Über die Gleichverteilung von Shnirel’man Density and Essential Components Zahlen mod. Eins, Math. Annalen 77 (1916), 313– The counting function A(x) of a set A of nonnegative 352, Satz 21. integers counts the number of positive integers in [Z] Antoni Zygmund, Trigonometric Series. I, II, A that do not exceed x, that is, Cambridge Univ. Press, 1959. A(x) 1. = a A Melvyn B. Nathanson 1 X2a x The Shnirel’man density of A is Paul Erd˝osand Additive Number Theory A(n) Additive Bases (A) inf . = n 1,2,... n Paul Erd˝os,while he was still in his twenties, = The sum of the sets A and B is the set A B wrote a series of extraordinarily beautiful papers + = a b : a A and b B . Shnirel’man proved the in additive and combinatorial number theory. The { + 2 2 } fundamental sumset inequality: key concept is additive basis. Let be a set of nonnegative integers, let be (A B) (A) (B) (A) (B). A h + + a positive integer, and let hA denote the set of This implies that if (A) > 0, then A is a basis of integers that can be represented as the sum of order h for some h. This does not apply directly exactly h elements of A, with repetitions allowed. to the sets of kth powers and the set of primes, A central problem in additive number theory is which have Shnirel’man density 0. However, it is to describe the sumset . The set is called hA A straightforward that if (A) 0 but (h0A) > 0 an additive basis of order if every nonnegative = h for some h0, then A is a basis of order h for some integer can be represented as the sum of exactly h. h elements of A. For example, the set of squares Landau conjectured the following strengthening is a basis of order 4 (Lagrange’s theorem), and of Shnirel’man’s addition theorem, which was the set of nonnegative cubes is a basis of order 9 proved by Mann [23] in 1942: (Wieferich’s theorem). (A B) min(1, (A) (B)). The set A of nonnegative integers is an as- + + ymptotic basis of order h if hA contains every Artin and Scherk [1] published a variant of Mann’s sufficiently large integer. For example, the set of proof, and Dyson [4], while an undergraduate at squares is an asymptotic basis of order 4 but Cambridge, generalized Mann’s inequality to h-fold not of order 3. The set of nonnegative cubes is sums. Nathanson [27] and Hegedüs, Piroska, and an asymptotic basis of order at most 7 (Linnik’s Ruzsa [17] have constructed examples to show theorem) and, by considering congruences modulo that the Shnirel’man density theorems of Mann 9, an asymptotic basis of order at least 4. The and Dyson are best possible. Goldbach conjecture implies that the set of primes We define the lower asymptotic density of a set is an asymptotic basis of order 3. Helfgott [18] A of nonnegative integers as follows: recently completed the proof of the ternary Gold- A(n) bach conjecture: Every odd integer n 7 is the dL(A) lim inf . = n 1,2,... n sum of three primes. = The modern theory of additive number theory be- This is a more natural density than Shnirel’man density. A set A with asymptotic density d (A) 0 gan with the work of Lev Genrikhovich Shnirel’man L = has Shnirel’man density (A) 0, but not con- (1905–1938). In an extraordinary paper [38] pub- = lished in Russian in 1930 and republished in an versely. A set A with asymptotic density dL(A) > 0 expanded form [39] in German in 1933, he proved is not necessarily an asymptotic basis of finite that every sufficiently large integer is the sum order, but A is an asymptotic basis if dL(A) > 0 and gcd(A) 1 (cf. Nash and Nathanson [24]). of a bounded number of primes. Not only did = Shnirel’man apply the Brun sieve, which Erd˝ossub- The set B of nonnegative integers is called an sequently developed into one of the most powerful essential component if tools in number theory, but he also introduced a (A B)> (A) + Melvyn B. Nathanson is professor of mathematics at for every set A such that 0 <
<<