On the size of finite Sidon sequences

Abstract

Let h ≥ 2 h \geq 2 be an integer. A set of positive integers B is called a B h {B_h} -sequence, or a Sidon sequence of order h, if all sums a 1 + a 2 + ⋯ + a h {a_1} + {a_2} + \cdots + {a_h} , where a i ∈ B ( i = 1 , 2 , … , h ) {a_i} \in B (i = 1,2, \ldots ,h) , are distinct up to rearrangements of the summands. Let F h ( n ) {F_h}(n) be the size of the maximum B h {B_h} -sequence contained in { 1 , 2 , … , n } \{ 1,2, \ldots ,n\} . We prove that \[ F 2 r − 1 ( n ) ≤ ( ( r ! ) 2 n ) 1 / ( 2 r − 1 ) + O ( n 1 / ( 4 r − 2 ) ) . {F_{2r - 1}}(n) \leq {({(r!)^2}n)^{1/(2r - 1)}} + O({n^{1/(4r - 2)}}). \]