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A note on sumsets of subgroups in $ℤ*_{p}$

100%

Hart D.

Acta Arithmetica

2013

tom 161

nr 4

387-395

Let A be a multiplicative subgroup of $ℤ*_p$. Define the k-fold sumset of A to be $kA = {x_1 + ... + x_k : x_i ∈ A, 1 ≤ i ≤ k}$. We show that $6A ⊇ ℤ*_p$ for $|A| > p^{11/23+ϵ}$. In addition, we extend a result of Shkredov to show that $|2A| ≫ |A|^{8/5-ϵ}$ for $|A| ≪ p^{5/9}$.

Waring's number for large subgroups of ℤₚ

51%

Cochrane T., Hart D., Pinner C., Spencer C.

Acta Arithmetica

2014

tom 163

nr 4

309-325

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t₄ = p^{22/39+ϵ}$, $t₅ = p^{15/29+ϵ}$ and for s ≥ 6, $tₛ = p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32} = p^{5/16+ϵ}$ and $t_{128} = p^{1/4}$.

Ograniczanie wyników

2 Acta Arithmetica

2 Hart D.

1 Cochrane T.

1 Pinner C.

1 Spencer C.

1 2014

1 2013

Wyniki wyszukiwania

A note on sumsets of subgroups in $ℤ*_{p}$

Waring's number for large subgroups of ℤₚ

Ograniczanie wyników

2 Acta Arithmetica

2 Hart D.

1 Cochrane T.

1 Pinner C.

1 Spencer C.

1 2014

1 2013

Wyniki wyszukiwania

A note on sumsets of subgroups in $ℤ*_{p}$

Waring's number for large subgroups of ℤ*ₚ*

Waring's number for large subgroups of ℤₚ