On additive decompositions of the set of primitive roots modulo p

Abstract

It is conjectured that the set G of the primitive roots modulo p has no decomposition (modulo p) of the form G = A+B with |A| > 2, |B| > 2. This conjecture seems to be beyond reach but it is shown that if such a decomposition of G exists at all, then |A|, |B| must be around p 1/2 , and then this result is applied to show that G has no decomposition of the form G = A + B + C with |A| > 2, |B| > 2, |C| > 2.

Additive Decompositions of Subgroups of Finite Fields

“…, m + n} ⊆ F p of n consecutive residues modulo p with two arbitrary sets A, B ⊆ F * p such that #A ≥ #B ≥ 2. Certainly various conjectures, similar to those of [6,17], can be about the non-existence of such decompositions. Here we merely show that some of the above methods apply to multiplicative decompositions too.…”

“…Let G d ⊆ F q be the group of dth powers. We start with the following generalisation of the bound (1), which closely follows the arguments of [6,17].…”

“…Sárközy [17] has conjectured that the set Q of quadratic residues modulo a prime p does not have nontrivial decompositions and showed towards this conjecture that any nontrivial decomposition Q = A + B satisfies (1) min{#A, #B} ≥ p 1/2 3 log p and (2) max{#A, #B} ≤ p 1/2 log p Furthermore, Dartyge and Sárközy [6] have made a similar conjecture for the set R of primitive roots modulo p and given the following analogues of (1) and (2): (3) min{#A, #B} ≥ ϕ(p − 1) τ (p − 1)p 1/2 log p and (4) max{#A, #B} ≤ τ (p − 1)p 1/2 log p where ϕ(k) and τ (k) are the Euler function and the number of integer positive divisors of an integer k ≥ 1. We also refer to [6,17] for further references about set decompositions.…”

Counting additive decompositions of quadratic residues in finite fields

“…Finally, we remark that the argument we use to prove Theorem 1 can be extended to prove results on additive decompositions of many other "multiplicatively" defined sets, such as cosets of multiplicative groups and sets of primitive elements of F * q . See [1,8] for analogues of (1) for such sets.…”

“…For any fixed ε > 0 there is a constant c > 0 such that for all integers k and m with q > k > q ε and q > m > q ε , we have N(k, m, q) ≤ cq1We fix a set V ⊆ F q of size #V = q ε/2 . We estimate the number N(V, k, m, q) of sets A, B ⊆ F q with #A = k, B = m such that Q = A + B and V ⊆ B.…”