Additive properties of random sequences of positive integers

Additive properties of random sequences of positive integers b P. ERDS and A. RÉNYI (Budapest) § 0. Introduction. It is well known (see e. g. [1]) that the number of those integers n < x which can be represente d in the form n = k 2+12 (k and l integers) has the order of magnitude x/logx ; as clearl the number of pairs k, l of positive integers such that k2 +12 < x is , =14, the reason wh the set of numbers which can be represented as the sum of two squares has ero densit is not that the squares are too rare, but-loosel speaking-that the are "too regularl" distributed, so that among the sums k2 +1 2 there are too man equal ones. This was first demonstrated b Atkin [2], who solved the following problem, proposed b J. E. Littlewood : If each square k2 is replaced b a random integer vk , chosen according to a certain probabilit law in the neighbourhood of k2, then the sums vk + vl almost surel have a positive densit. In § 1 of the present paper we introduce a class of sequences of random integers. This construction has been used alread in [3]. In § 2, 3 and 4 we prove some theorems of a similar character to that of Atkin, mentioned above. We shall show that if the random sequences wk an(. µ, have approximatel the same order of magnitude as the sequence ck2 with some c > 0, then the sequences vk+,a,, V+,u i and vj,+vl will have positive densit with probabilit 1 ; moreover, in all three cases the sequences of numbers n which have exactl r representations in the form n = vk + µi , n = k 2+p, or n = vk +v, (k < l), will almost surel have a positive densit for each value of r (r = 0, 1,. . .) and these densities form a Poisson distribution. In § ó we shall show that the number f(n) of representations of n in the form n = vk + vi has, in case it tends to + oo, a normal distribution in the limit. In § 6 we generalie these results for sums of more than two terms of a random sequence of integers. In § 7 we consider the distribution of differences of a random sequence. § 8 …