[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 λn (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 σ(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 σ(A B) σ(A) σ(B) σ(A)σ(B). A h + ≥ + − a positive integer, and let hA denote the set of This implies that if σ(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 σ(A) 0 but σ(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). σ(A B) min(1,σ(A) σ(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 σ(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 σ(A B)>σ(A) + Melvyn B. Nathanson is professor of mathematics at for every set A such that 0 <σ(A) < 1. Lehman College (CUNY). His email address is melvyn. Shnirel’man’s inequality implies that every set [email protected]. of positive Shnirel’man density is an essential 140 Notices of the AMS Volume 62, Number 2 component. There exist sparse sets of zero asymp- totic density that are not essential components. Khinchin [20] proved that the set of nonnegative squares is an essential component. Note that the set of squares is a basis of order 4. Using an extremely clever elementary argument, Erd˝os[6], at the age of twenty-two, proved the following considerable improvement: Every additive basis is an essential component. Greatly impressed, Lan- dau celebrated this result in his 1937 Cambridge Tract Über einige neuere Fortschritte der additiven Zahlentheorie [22]. Photo courtesy of Melvyn Nathanson Plünnecke [33], [34], [35] and Ruzsa [37] have Mel Nathanson with Paul Erd˝os,who is holding made important contributions to the study of Mel’s infant son, Alex, in 1988. essential components. exist, but I was able to construct asymptotic bases The Erd˝os-TuránConjecture of order 2 that were both thin and minimal [25]. In another classic paper, published in 1941, Erd˝os Of course, none was a counterexample to the and P. Turán [5] investigated Sidon sets. The set Erd˝os-Turánconjecture. A of nonnegative integers is a Sidon set if every Stöhr [41] gave the first definition of minimal integer has at most one representation as the sum asymptotic basis, and Härtter [16] gave a non- of two elements of A. They concluded their paper constructive proof that there exist uncountably as follows: many minimal asymptotic bases of order h for Let f (n) denote the number of representa- every h 2. ≥ tions of n as a a , … . If f (n) > 0 for n>n0, There is a natural dual to the concept of i + j then lim sup f (n) . Here we may mention a minimal asymptotic basis. We call a set A =1 that the corresponding result for g(n), the an asymptotic nonbasis of order h if is not an number of representations of n as aiaj , can asymptotic basis of order h, that is, if there are be proved. infinitely many positive integers not contained in The additive statement is still a mystery. The the sumset hA. An asymptotic nonbasis of order h is maximal if A b is an asymptotic basis Erd˝os-Turánconjecture, that the representation [{ } function of an asymptotic basis of order 2 is of order h for every nonnegative integer b ∉ A. always unbounded, is a major unsolved problem The set of even nonnegative integers is a trivial in additive number theory. example of a maximal nonbasis of order h for every h 2, and one can construct many other Many years later, in 1964, Erd˝os[7] published ≥ the proof of the multiplicative statement. This examples that are unions of the nonnegative parts proof was later simplified by Ne˘set˘riland Rödl [32] of congruence classes. It is difficult to construct and generalized by Nathanson [26]. nontrivial examples. Long ago, while a graduate student, I searched for I discussed this and other open problems in my a counterexample to the Erd˝os-Turánconjecture. first paper [25] in additive number theory. I did not Such a counterexample might be extremal in know Erd˝osat the time, but I mailed him a preprint several ways. It might be “thin” in the sense that of the article. It still amazes me that he actually it contains few elements. Every asymptotic basis read this paper sent to him out of the blue by a of order h has counting function A(x) cx1/h for completely obscure student, and he answered with ≥ some c>0 and all sufficiently large x. We call a long letter in which he discussed his ideas about 1/h one of the problems. This led to correspondence, an additive basis of order h thin if A(x) c0x meetings, and joint work over several decades. for some c0 > 0 and all sufficiently large x. Thin bases exist. The first examples were constructed in the 1930s by Raikov [36] and by Stöhr [40], and Extremal Properties of Bases Cassels [2], [29] later produced another important Here is a small sample of results on minimal bases class of examples. and maximal nonbases. Alternatively, an asymptotic basis A of order Nathanson and Sarközy [31] proved that if A is h might be extremal in the sense that no proper a minimal asymptotic basis of order h 2, then ≥ subset of A is an asymptotic basis of order h. This d (A) 1/h. The proof uses Kneser’s theorem [21] L means that removing any element of A destroys on the asymptotic density of sumsets, one of the every representation of infinitely many integers.