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On an additive representation function

JOURNAL OF NUMBER THEORY 2004年第2期104卷 327-334页

作者： Balasubramanian, R Prakash, G Inst Math Sci Madras 600113 Tamil Nadu India

Let A be an infinite subset of natural numbers, n is an element of N and X a positive real number. Let r(n) denotes the number of solution of the equation n = a(1) + a(2) where a(1) less than or equal toa(2) and a(1), a(2) is an element of A. Also let \A(X)\ denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)not equal1, we improve the lower bound of \A (X)\ given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible. (C) 2003 Elsevier Inc. All rights reserved.

关键词： addition of sequence additive representation function

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Generalizations of some results about the regularity properties of an additive representation function

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ACTA MATHEMATICA HUNGARICA 2019年第1期157卷 121-140页

作者： Kiss, S. Z. Sandor, Cs. Budapest Univ Technol & Econ Inst Math H-1529 Budapest Hungary

Let A={a1,a2,}(a1B(A,N)then |1(RA,2(n))| cannot be bounded, where B(A,N) denotes the number of blocks formed by consecutive integers in A up to N and l denotes the l-th difference. Their result was extended to l(RA,2(... 详细信息

关键词： additive number theory general sequence additive representation function

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On additive representation function of general sequences

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ACTA MATHEMATICA HUNGARICA 2007年第1-2期115卷 169-175页

作者： Horvath, G. Coll Dunaujvaros Inst Math Dept Math Anal H-2401 Dunaujvaros Hungary

Let 0 0 is an integer, then there does not exist n(0) such that d n(0).

Let 0 <= a(1) < a(2) < center dot center dot center dot be an infinite sequence of integers and let r(1) (A, n) = vertical bar(i;j) : a(i) + a(j) = n, i <= j vertical bar. We show that if d > 0 is an integer, then there does not exist n(0) such that d <= r(1) (A, n) <= d + [root 2d+1/2] for n > n(0).

关键词： additive number theory general sequences additive representation function

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On Bh-sets which are asymptotic bases of order 2h

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JOURNAL OF NUMBER THEORY 2025年 266卷 350-376页

作者： Kiss, Sandor Z. Sandor, Csaba Budapest Univ Technol & Econ Inst Math Dept Algebra & Geometry Muegyet Rkp 3 H-1111 Budapest Hungary HUN REN BME Stochast Res Grp Muegyet Rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Inst Math Dept Stochast Muegyet Rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Dept Comp Sci & Informat Theory Muegyet Rkp 3 H-1111 Budapest Hungary MTA BME Lendulet Arithmet Combinator Res Grp Muegyet Rkp 3 H-1111 Budapest Hungary

Let h, k >= 2be integers. A set Aof positive integers is called asymptotic basis of order kif every large enough positive integer can be written as the sum of k terms from A. A set of positive integers Ais said to be a B-h[g]-set if every positive integer can be written as the sum of h terms from A at most g different ways. In this paper we prove the existence of B-h[1] sets which are asymptotic bases of order 2h by using probabilistic methods. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// ***/licenses/by/4.0/).

关键词： additive number theory General sequences additive representation function Sidon sets

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Unique representations of integers by linear forms

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2025年 213卷

作者： Kiss, Sandor Z. Sandor, Csaba Budapest Univ Technol & Econ Inst Math Dept Algebra & Geometry Muegyet Rkp 3 H-1111 Budapest Hungary HUN REN BME Stochast Res Grp Muegyet rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Inst Math Dept Stochast Muegyet Rkp 3 H-1111 Budapest Hungary HUN REN Alfred Reny Inst Math Realtanoda Utca 13-15 H-1053 Budapest Hungary MTA HUN REN RI Lendulet Momentum Arithmet Combinat Realtanoda utca 13-15 H-1053 Budapest Hungary

Let k >= 2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers lambda_=(lambda(1),& mldr;,lambda(k)) with 1 = 0.

Let k >= 2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers lambda_=(lambda(1),& mldr;,lambda(k)) with 1 <=lambda(1)additive representation function R-A,R-lambda_(n)=|{(a(1),& mldr;,a(k))is an element of A(k):lambda(1)a(1)+& ctdot;+lambda(k)a(k)=n}|. For k=2, Moser constructed a set A of nonnegative integers such that R-A,R-lambda_(n)=1 holds for every nonnegative integer n. In this paper we characterize all the k-tuples lambda_ and the sets A of nonnegative integers with R-A,R-lambda_(n)=1 for every integer n >= 0.

关键词： additive number theory additive representation function Linear forms

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additive representation functionS AND DISCRETE CONVOLUTIONS

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ACTA MATHEMATICA HUNGARICA 2022年第2期166卷 217-238页

作者： Sandor, Cs Budapest Univ Technol & Econ Inst Math Dept Stochast Muegyet Rkp 3 H-1111 Budapest Hungary MTA BME Lendulet Arithmet Combinator Res Grp H-1529 Budapest Hungary

For a set A of non-negative integers, let R-A(n) denote the number of solutions to the equation n = a + a' with a, a' is an element of A. Denote by chi(A)(n) the characteristic function of A. Let b(n) > 0 be a sequence satisfying lim sup(n ->infinity) b(n) < 1. In this paper, we prove some Erdos-Fuchs-type theorems about the error terms appearing in the approximation formulae for R-A(n) = Sigma(n)(k=)(0) chi(A)(k)chi(A)(n - k) and Sigma(N)(n=0) R-A(n) having principal terms Sigma(n)(k=)(0) b(k)b(n-k) and Sigma(N)(n=0) Sigma(n)(k=)(0) b(k)b(n-k), respectively.

关键词： additive representation function Erdos-Fuchs theorem

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ON additive representation functionS OF FINITE SETS, I (VARIATION)

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PERIODICA MATHEMATICA HUNGARICA 2013年第2期66卷 201-210页

作者： Sarkoezy, Andras Eotvos Lorand Univ Dept Algebra & Number Theory H-1117 Budapest Hungary

If m is an element of N, Z(m) is the additive group of the modulo m residue classes, A subset of Z(m) and n is an element of Z(m), then let R(A, n) denote the number of solutions of a + a' = n with a, a' is an element of A. The variation V (A) = (n is an element of Zm)max vertical bar R(A, n + 1) - R(A, n)vertical bar is estimated in terms of the number of a's with a - 1 is not an element of A, a is an element of A.

关键词： additive representation function variation

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On a regularity property of additive representation functions

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Periodica Mathematica Hungarica 2005年第2期51卷 31-35页

作者： Kiss, Sándor Department of Algebra and Number Theory Eötvös Loránd University H-1117 Budapest Pazmany Peter Setany 1/c Hungary

Let A = {a 1, a 2, . . .}(a 1 ≤ a 2 ≤ . . .) be an infinite sequence of nonnegative integers, and let R(n) denote the number of solutions of a x + a y = n (a x, a y ∈ A). P. Erdos, A. Sárközy and V. T. S&... 详细信息

Let A = {a ₁, a ₂, . . .}(a ₁ ≤ a ₂ ≤ . . .) be an infinite sequence of nonnegative integers, and let R(n) denote the number of solutions of a _x + a _y = n (a _x, a _y ∈ A). P. Erdos, A. Sárközy and V. T. Sós proved that if lim _N→∞ B(A,N)/√N = +∞ then |Δ ₁(R(n))| cannot be bounded, where B(A, N) denotes the number of blocks formed by consecutive integers in A up to N and Δ _k denotes the k-th difference. The aim of this paper is to extend this result toΔ _k(R(n)) for any fixed k ≥ 2. © Akadémiai Kiadó, Budapest.

关键词： additive number theory additive representation function General sequences k-th difference

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ON MONOTONE INCREASING representation functionS

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BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2024年第2期109卷 196-205页

作者： Kiss, Sandor Z. Sandor, Csaba Yang, Quan-hui Budapest Univ Technol & Econ Inst Math Muegyetem Rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Dept Comp Sci & Informat Theory Muegyetem Rkp 3 H-1111 Budapest Hungary MTA BME Lendulet Arithmet Combinator Res Grp ELKH Muegyetem Rkp 3 H-1111 Budapest Hungary Nanjing Univ Informat Sci & Technol Sch Math & Stat Nanjing 210044 Peoples R China

Let k >= 2 be an integer and let A be a set of nonnegative integers. The representation function R-A,R-k(n) for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let A(n) denote the counting function of A. Bell and Shallit ['Counterexamples to a conjecture of Dombi in additive number theory', Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if A(n) = o(n((k-2)/k-epsilon)) for some e > 0, then R-N\A,R-k(n) is eventually strictly increasing. We improve this result to A(n) = O(n((k-2)/(k-1))). We also give an example to show that this bound is best possible.

关键词： additive number theory additive representation function monotonicity

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Generalized Sidon sets of perfect powers

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RAMANUJAN JOURNAL 2022年第2期59卷 351-363页

作者： Kiss, Sandor Z. Sandor, Csaba Budapest Univ Technol & Econ Inst Math Dept Algebra Muegyet Rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Inst Math Dept Stochast Muegyet Rkp 3 H-1111 Budapest Hungary Budapest Univ Technol & Econ Dept Comp Sci & Informat Theory Muegyet Rkp 3 H-1111 Budapest Hungary ELKH MTA BME Lendulet Arithmet Combinator Res Grp Muegyet Rkp 3 H-1111 Budapest Hungary

For h >= 2 and an infinite set of positive integers A, let R-A,R-h(n) denote the number of representations of the positive integer n as the sum of h distinct terms from A. A set of positive integers A is called a B-h[g] set if every positive integer can be written as the sum of h not necessarily distinct terms from A at most g different ways. We say a set A is a basis of order h if every positive integer can be represented as the sum of h terms from A. Recently, Vu [17] proved the existence of a thin basis of order h formed by perfect powers. In this paper, we study weak B-h[g] sets formed by perfect powers. In particular, we prove the existence of a set A formed by perfect powers with almost possible maximal density such that R-A,R-h(n) is bounded by using probabilistic methods.

关键词： additive number theory General sequences additive representation function Sidon set

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