On additive representation functions

R. Balasubramanian; Sumit Giri

doi:10.1142/S1793042115500633

On additive representation functions

R. Balasubramanian, Sumit Giri

Research output: Contribution to journal › Article › peer-review

Abstract

Let = {a₁ < a₂ < a₃ < ⋯ < a_n < ⋯} be an infinite sequence of nonnegative integers and let R₂(n) = |{(i, j): a_i + a_j = n; a_i, a_j ; i ≤ j}|. We define S_k = ∑l = 1^k(R₂(2l)-R₂(2l + 1)). We prove that if the L^∞-norm of S_k⁺(= max S_k, 0) is small, then the L¹-norm of 1/4S_k⁺k is large.

Original language	English
Pages (from-to)	1165-1176
Number of pages	12
Journal	International Journal of Number Theory
Volume	11
Issue number	4
DOIs	https://doi.org/10.1142/S1793042115500633
State	Published - 5 Jun 2015
Externally published	Yes

Keywords

Additive representation functions
Sidon set
general sequence
monotonicity

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title = "On additive representation functions",

abstract = "Let = {a1 < a2 < a3 < ⋯ < an < ⋯} be an infinite sequence of nonnegative integers and let R2(n) = |{(i, j): ai + aj = n; ai, aj ; i ≤ j}|. We define Sk = ∑l = 1k(R2(2l)-R2(2l + 1)). We prove that if the L∞-norm of Sk+(= max Sk, 0) is small, then the L1-norm of 1/4Sk+k is large.",

keywords = "Additive representation functions, Sidon set, general sequence, monotonicity",

author = "R. Balasubramanian and Sumit Giri",

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AU - Balasubramanian, R.

AU - Giri, Sumit

PY - 2015/6/5

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N2 - Let = {a1 < a2 < a3 < ⋯ < an < ⋯} be an infinite sequence of nonnegative integers and let R2(n) = |{(i, j): ai + aj = n; ai, aj ; i ≤ j}|. We define Sk = ∑l = 1k(R2(2l)-R2(2l + 1)). We prove that if the L∞-norm of Sk+(= max Sk, 0) is small, then the L1-norm of 1/4Sk+k is large.

AB - Let = {a1 < a2 < a3 < ⋯ < an < ⋯} be an infinite sequence of nonnegative integers and let R2(n) = |{(i, j): ai + aj = n; ai, aj ; i ≤ j}|. We define Sk = ∑l = 1k(R2(2l)-R2(2l + 1)). We prove that if the L∞-norm of Sk+(= max Sk, 0) is small, then the L1-norm of 1/4Sk+k is large.

KW - Additive representation functions

KW - Sidon set

KW - general sequence

KW - monotonicity

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JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 4

ER -

On additive representation functions

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Keywords

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