On arithmetic properties of sumsets

A. Balog (1) , Joel Rivat (2) , András Sárközy (3)

1 Alfréd Rényi Institute of Mathematics
2 I2M - Institut de Mathématiques de Marseille
3 Department of Algebra and Number Theory [Budapest]

Let \({\mathcal{A}}\), \({\mathcal{B}}\) be large subsets of \({\{1,\ldots,N\}}\). Arithmetic properties of the sums a + b with \({a\in\mathcal{A}}\), \({b\in\mathcal{B}}\) are studied. In particular, the existence of sums a + b with a prime divisor belonging to a large set \({\mathcal{P}}\) of primes is proved, min a,b ω(a + b) is studied, and the solvability of both f(a + b) = +1 and −1 is shown for multiplicative arithmetic functions f(n) satisfying certain conditions. Two related open problems are also presented.

Mots clés: sumset; large prime factor; number of prime factors; multiplicative arithmetic function

Domaines: Théorie des nombres [math.NT]

A. Balog, Joel Rivat, András Sárközy. On arithmetic properties of sumsets. Acta Mathematica Academiae Scientiarum Hungaricae, 2014, 144 (1), pp.18-42. ⟨10.1007/s10474-014-0436-y⟩. ⟨hal-01260090⟩