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Let $$G$$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $$c$$ such that whenever $$S$$ is a non-trivial normal subset in $$G$$ then $$S^{k} = G$$ for any integer $$k$$ at least $$c \cdot (\log|G|/\log|S|)$$ . This result is generalized by showing that there exists an absolute constant $$c$$ such that whenever $$S_{1}$$ , $$\ldots , $$ $$S_{k}$$ are normal subsets in $$G$$ with $$\prod_{i=1}^{k} |S_{i}| \geq {|G|}^{c}$$ then $$S_{1} \cdots S_{k} = G$$ .

An example is presented of a subset $A$ of $\mathbb Z_N$ of density $\alpha$ such that the largest non-trivial Fourier coefficient of the characteristic function of $A$ is very small, but the probability that a random arithmetic progression (mod $N$) of length 4 lies in $A$ is significantly smaller than $\alpha^4$., Comment: This preprint was written well over a decade ago and has been known about by experts but not made publicly available. I apologize for this and am now remedying the situation. Revised version updates the notation, corrects minor errors (pointed out by referee), and adds one or two references

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors $$p_{1},\ldots, p_{t}$$ of n and positive integers $$\alpha_{1}, \ldots , \alpha_{t}$$ such that $$n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}$$. Erdős and Hegyvari [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible by p. Recently, Fang and Chen [3] proved that for any given positive integer m, there are infinitely many weakly prime-additive numbers which are divisible bym with t = 3 if and only if $$8 \nmid m$$. In this paper, we prove that for any given positive integer m, the number of weakly prime-additive numbers which are divisible by m and less than x is larger than $${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$$ for all sufficiently large x, where c is a positive absolute constant. The constant c depends on the result on the least prime number in an arithmetic progression.

Let \({\mathcal {L}}\) be the set of integers n which can be written as $$ n=p_1^3 + p_2^3+p_3^3+p_4^3. $$ Using the circle method and sieves, we prove that \({\sum_{N 0}\) is an absolute constant.

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then \(|A|\leqq 2^{-ck/{(\log k)}^{2}}N\), for some absolute constant c>0, provided that N is large enough.