library/additive_bases/sarkozy_1997_additive_representation_functions

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András Sárközy, Vera T. Sós, On Additive Representation Functions. The Mathematics of Paul Erdős I (R. L. Graham et al., eds.), Springer, 1997, 129-150. doi:10.1007/978-3-642-60408-9_11.

This is a survey chapter on the additive representation functions r_1, r_2, r_3 of a set A of nonnegative integers, covering their regularity properties and value distribution, with a few new results and many open problems. Section 2 fixes notation and defines the classes B_2[g] of sets in which every n has at most g representations n = a + a' with a <= a', the case g = 1 being Sidon sets. Section 3 treats representation functions of general sequences via the Erdos-Turan (1941) and Dirac-Newman theorems, which show r_1(A,n) and r_2(A,n) cannot be eventually constant for an infinite A, and presents the short generating-function proof; the Erdos-Fuchs theorem and its relatives follow. The methods surveyed are exponential sums, combinatorial counting, and Erdos-Renyi probabilistic arguments. For problem 156 the chapter is the freely available first-edition predecessor of the revised 2013 edition, which is not held here: it defines finite maximal Sidon sets and remarks that little is known about their cardinalities, but the question it poses concerns B_2[g] embeddings rather than the maximal-Sidon frontier, and it predates Ruzsa's (N log N)^{1/3} bound, so it is off-point for the exact frontier.

Source: https://real.mtak.hu/110610/.

Statements recorded.

• Section 2 definitions: Defines the representation functions r_1, r_2, r_3 of A and the classes B_2[g] of sets where a + a' = n with a <= a' has at most g solutions; B_2[1] are the Sidon sets.
• Erdos-Turan (1941), quoted: For an infinite set A in N, r_1(A,n) cannot be constant from some point on.
• Dirac-Newman, quoted with proof: The same non-eventual-constancy holds for r_2(A,n), proved by a short generating-function argument with f(x) the sum of x^a over a in A.