On the subset sums of exponential type sequences

• Autorzy: Yong-Gao Chen , Jin-Hui Fang , Norbert Hegyvári
• Języki publikacji: EN

Abstrakty

EN

For a sequence A ⊆ ℕ, let P(A) be the set of all sums of distinct terms taken from A. The sequence A is said to be complete if P(A) contains all sufficiently large integers. Let p > 1 be an integer. The following main results are proved: (a) Let $A_t = {a_1 ≤ ... ≤ a_t}$ be any sequence of positive integers (not necessarily distinct), $S_p = {p^i : i = 0, 1, ... }$ and $S_p A_t = {p^i a_j : i = 0, 1, ...; j = 1, ..., t}$. When t ≥ p-1, the sequence P(S_pA_t)$ has positive lower asymptotic density not less than $1/a_{p-1}$. The lower bounds p-1 and $1/a_{p-1}$ are both the best possible. (b) For any positive integer k, the sequence ${ p^i F_j : i = 0, 1, ... ; j = k, k+1, ..., n}$ is complete, where $F_j$ is the jth Fibonacci number and $n = p^2 F_{k+2p-1}^2$.

Kategorie tematyczne

• 11B13: Additive bases, including sumsets
• 11A07: Congruences; primitive roots; residue systems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 173
• Numer: 2
• Strony: 141-150

Daty

wydano

2016

Twórcy

autor

Yong-Gao Chen

• School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, P.R. China

autor

Jin-Hui Fang

• Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, P.R. China

autor

Norbert Hegyvári

• ELTE TTK, Eötvös University, Institute of Mathematics, Pázmány St. 1/c, H-1117 Budapest, Hungary

Identyfikatory

DOI

10.4064/aa8133-3-2016