We prove that a multiplicative subgroup Ak of Z∗p is a generalized arithmetic progression if and only if |Ak|=2,4, or p−1. Much of the argument builds upon recent work studying additive decompositions of subgroups, and we generalize a result of Hanson and Petridis to show that any additive n-decomposition of a subgroup must be a direct sum. We also show how this classification quickly follows from Kalmynin’s recent work resolving Sárközy’s conjecture for quadratic residues.
Let P denote the set of all primes, and let delta(P) denote the relative lower density of a subset P of P. Suppose that P1 and P2 are two subsets of P satisfying delta(P)1 > 0 and delta(P)2 > root 2/2. Then for every sufficiently large integer n equivalent to 5 (mod 24), there exist p(1) is an element of P-1, p(2) is an element of P-2, and p(3), p(4),p(5) is an element of P such that n = p(1)(2) + p(2)(2) + p(3)(2) + p(4)(2) + p(5)(2). Moreover, we also provide two more density versions of Hua's theorem and a square-free version of the Cauchy-Davenport type inequality.
We correct a gap in the paper of the title. Let f(T) be a monic polynomial of degree d with coefficients in a finite field Fq. We give a criterion for f to satisfy the following property: for all but d2−d−1 values of s in Fq, the probability that f(T)+sT+b has a given factorization type λ over Fq, as b∈Fq is chosen uniformly at random, is pλ+Od(q−1/2), where pλ is the probability that a permutation in Sd has cycle shape λ. The assumptions allow (q,2d)>1.
Let C and W be two sets of integers. If C+W=Z, then we say that C is an additive complement to W. If no proper subset of C is an additive complement to W, then we say that C is a minimal additive complement to W. Let m and h be two positive integers with m≥2, h≡1(modm). We show that there exists an infinite, not eventually periodic set W={wi}∞i=1 such that wi+1−wi∈{m,h} for all i and W admits a minimal additive complement.
Let p be a prime number. We denote by s & lowast;p the smallest integer l such that, out of any given l integers coprime with p, one can select p integers such that the sum of the p integers is a multiple of p, but not a multiple of p2. It is conjectured that s & lowast;p = 2p + 1 for any prime number p >= 3. We give a non-trivial upper bound s & lowast;p <= 3p-2.
