On Additive Representative Functions

Abstract

In this paper we give a short survey of additive representation functions, in particular, on their regularity properties and value distribution. We prove a couple of new results and present many related unsolved problems.

1 Appendix

A1. Introduction

The paper above appeared in 1997. Since that time more than 100 papers have been published on related problems. In this Appendix our goal is to give a short survey of these papers. In order to limit the extent of it we will focus on the most important results, and in the reference list we will present only the records of the most important and most recent papers, and a few survey papers; the references to the further related work can be found in these papers.

A2. Notations

We will keep the notations and the reference numbers of the original paper; thus, e.g., Problem 2 will refer to the second problem in Sect. 3 of the original paper. On the other hand, we will refer to the sections and references in the Appendix by using a prefix A so that, e.g., the second item in the reference list of the Appendix is marked as [43].

A3. The Representation Function of General Sequences. The Erdős-Fuchs Theorem and Related Results

Sárközy [88] proved the following local version of Theorem 1 of Erdős and Fuchs: for all C > 0 there are N ₀ = N ₀(C) and C ₁ = C ₁(C) so that if $\mathcal{A}\subset \mathbb{N}$ and N > N ₀, then there is an M with

$$\displaystyle{N < M \leq {N}^{2}\mbox{ and }\sum _{ n=1}^{M}{(R(n) - C)}^{2} >C_{ 1}M.}$$

He also showed that this result is best possible: for all $\varepsilon > 0$ there is an $\mathcal{A}\subset \mathbb{N}$ such that for infinitely many N we have

$$\displaystyle{\sum _{n=1}^{M}{(R(n) - 2)}^{2} <\varepsilon M\mbox{ for all }N < M < \frac{\varepsilon } {136}{N}^{2}.}$$

Ruzsa [81] proved a “converse” of the Erdős-Fuchs theorem (Theorem 2) by showing that there exists a non-decreasing sequence $\mathcal{A}$ of nonnegative integers such that

$$\displaystyle{\sum _{n=1}^{N}r_{ 1}(\mathcal{A},n) = cN + O({N}^{1/4}\log N)}$$

for some constant c > 0.

Tang [93] sharpened Vaughan’s result [40] on the extension of the Erdős-Fuchs theorem to k term sums, and later Chen and Tang [46] estimated the constant implied by the ordo notation.

Horváth [68] extended the Erdős-Fuchs theorem further by considering the sum $\mathcal{A}_{1} + \mathcal{A}_{2} + \cdots + \mathcal{A}_{k}$ of different sets $\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{k}$, and later in another paper [64] he sharpened this result.

Let $\mathcal{A} =\{ a_{1} \leq a_{2} \leq \cdots \,\}$ be an infinite sequence of nonnegative integers, and write

$$\displaystyle{R(\mathcal{A},x;k) = \left \vert \left \{{\bigl (a_{i_{1}},\ldots,a_{i_{k}}\bigr )} \in {\mathcal{A}}^{k} : a_{ i_{1}} + \cdots + +a_{i_{k}} \leq x\right \}\right \vert }$$

and

$$\displaystyle{P(\mathcal{A},x;k) = R(\mathcal{A},x;k) - cx.}$$

Chen and Tang [45] estimated the mean square of this discrepancy $P(\mathcal{A},x;k)$.

Lev and Sárközy [74] proved an Erdős-Fuchs-type theorem for finite groups, and they showed that their result is sharp.

Horváth [65] proved the following theorem which is closely related to the first theorem of Erdős and Fuchs (Theorem 1): If $\mathcal{A} =\{ a_{1},a_{2},\ldots \}$ $(a_{1} < a_{2} < \cdots \,)$ is an infinite sequence of nonnegative integers and d is a positive integer then there is no integer n ₀ such that for all n > n ₀ we have

$$\displaystyle{d \leq r_{3}(\mathcal{A},n) \leq d + \left [\sqrt{2d} + \frac{1} {2}\right ].}$$

Sándor [86] proved a similar theorem, and Chen and Tang [49] extended Horváth’s theorem to k term sums and the k term analogues of the other two functions r ₁ and r ₂.

In our original paper we mentioned the results of Erdős and Sárközy [14, 15] that if the function f(n) satisfies certain assumptions, then (3) cannot hold, and that this theorem is nearly sharp. Horváth [66] extended the first result to k term sums in place of $r_{1}(\mathcal{A},n)$, and Kiss [71] proved that Horváth’s result is nearly best possible.

In [16] Erdős, Sárközy and T. Sós proved that if $\mathcal{A}$ is an infinite set of positive integers, and, denoting the number of blocks formed by consecutive integers in $\mathcal{A}$ up to N by $B(\mathcal{A},N)$, we have

$$\displaystyle{\lim _{N\rightarrow +\infty }\frac{B(\mathcal{A},N)} {{N}^{1/2}} = +\infty,}$$

then the differences $\left \vert r_{1}(\mathcal{A},n + 1) - r_{1}(\mathcal{A},n)\right \vert$ cannot be bounded. They also showed that this result is best possible. Kiss extended the theorem to kth differences $\left \vert \Delta _{k}(R(n))\right \vert$, and later he also showed [69] that his result is sharp.

In a recent paper Sárközy [89] studied the analogues in $\mathbb{Z}/m\mathbb{Z}$ of the problems considered in [16].

The results of Erdős, Sárközy and T. Sós [17, 18], resp. Balasubramanian [2] on the monotonicity properties of additive representation functions have been extended by Tang [94], Chen and Tang [47, 48], resp. Chen, Sárközy, T. Sós and Tang [50] in various directions. In particular, it is proved in [48] and [50] that if $\mathcal{A}$ is an infinite set of positive integers such that its complement $\mathcal{B} = \mathbb{N} \setminus \mathcal{A}$ satisfies certain simple conditions then $r_{2}(\mathcal{A},n)$ cannot be ultimately increasing. However, Problem 1 is still open in its original form.

S. Giri settled the first half of Problem 2 by constructing a set $\mathcal{A}$ of the desired properties (unpublished yet). It might be interesting to study the second half of the problem as well: how dense can $\mathbb{N} \setminus \mathcal{A}$ be for such a set $\mathcal{A}$?

Problems 3–5 are still open.

A4. A Conjecture of Erdős and Turán and Related Problems and Results

Grekos, Haddad, Helou and Pihko [60] proved that if $\mathcal{A}$ is a set of nonnegative integers such that

$$\displaystyle{ r_{1}(\mathcal{A},n) \geq 1 }$$

(A4.1)

for every $n \in \mathbb{N}$ then we have $r_{1}(\mathcal{A},n) > 5$ for infinitely many n, and Borwein, Choi and Chu improved this to $r_{1}(\mathcal{A},n) > 7$.

Konstantoulas [72] proved that if there is a number n ₀ such that if (A4.1) holds for n > n ₀ then we have $r_{1}(\mathcal{A},n) > 5$ for infinitely many n.

By Ruzsa’s Theorem 4 there exists an asymptotic basis $\mathcal{A}$ of order 2 such that for N > N ₀ we have

$$\displaystyle{ \frac{1} {N}{\biggl (\sum _{n=1}^{N}r_{ 1}^{2}(\mathcal{A},n)\biggr )} < C}$$

for some absolute constant C. In two papers Tang [92] presented explicit values for these constants N ₀, C.

For $m \in \mathbb{N}$ let R _m denote the least integer such that there is a set $\mathcal{A}\subset \mathbb{Z}/m\mathbb{Z}$ with $\mathcal{A} + \mathcal{A} = \mathbb{Z}/m\mathbb{Z}$ and ${\bigl |\{(a,b) : a + b = n,\ a,b \in \mathcal{A}\}\bigr |}\leq R_{m}$ for all $n \in \mathbb{Z}/m\mathbb{Z}$. It follows from Ruzsa’s result above that R _m is bounded. Chen [44] proved the uniform bound R _m  ≤ 288, and Chen and Tang gave better bound for certain m values of special form.

Konyagin and Lev [73] studied and settled the Erdős-Turán problem in infinite Abelian groups. They determined what are the infinite Abelian groups G for which the analogue of the Erdős-Turán conjecture holds and what are the ones for which it fails, and in both cases they provide further information on the number of representations of the elements g of G in the form $a + a^{\prime} = g$ with a, a′ belonging to a basis $\mathcal{A}$ of G.

(See also a paper of Haddad and Helou [62].)

In Sect. 4 we mentioned the conjecture of Erdős and Freud that if $\mathcal{A}\subset \mathbb{N}$ is infinite and $r_{2}(\mathcal{A},n)$ is bounded then there are infinitely many n with

$$\displaystyle{ r_{2}(\mathcal{A},n) = 1, }$$

(A4.2)

and probably there are more integers n satisfying (A4.2) than integers n with

$$\displaystyle{r_{2}(\mathcal{A},n) > 1.}$$

Our Theorem 5 above disproved this second stronger version of the conjecture of Erdős and Freud. Sándor [87] also disproved the weaker version of the conjecture by constructing an infinite set $\mathcal{A}$ of nonnegative integers for which $r_{2}(\mathcal{A},n) \leq 3$ for all n and it assumes only the values 0, 2 and 3 infinitely many times. Sándor’s construction also disproves the conjecture formulated in our Problem 6 but it does not settle Problem 7. Moreover, in Sándor’s construction the counting function A(n) of $\mathcal{A}$ grows slowly:$A(n) = O{\bigl ({(\log n)}^{2}\bigr )}$ $A(n) = O{\bigl ({(\log n)}^{2}\bigr )}$. Thus it remains to see whether there exists a set $\mathcal{A}$ such that A(n) ≫ n ^c for some c > 0 and all n, $r_{2}(\mathcal{A},n)$ is bounded, and (A4.2) has only finitely many solutions.

A5. Sidon Sets: The Erdős-Turán Theorem, Related Problems and Results

This has been a very intensively studied field in the last 15 years. Since the extent of this Appendix is limited thus we have to restrict ourselves to listing some of the most important papers written on this subject. If the reader wants to know more on the papers written on Sidon sets, then O’Bryant’s excellent survey paper [77] can be used, while for more information on large B _h [g] sets one should consult the paper of Cilleruelo, Ruzsa and Vinuesa [51].

In our original paper we mentioned the result of Ajtai, Komlós and Szemerédi [1] on dense infinite Sidon sets: they proved that there is an infinite Sidon set $\mathcal{A}$ with A(n) ≫ (nlogn)^1 ∕ 2. Ruzsa [83] improved on this significantly by proving that there is an infinite Sidon set $\mathcal{A}$ with $A(n) = {n}^{\sqrt{2}-1+o(1)}$.

Ruzsa [84] showed that there is a maximal Sidon set $\mathcal{A}\subset \{ 1,2,\ldots,N\}$ with $\vert \mathcal{A}\vert \ll {(N\log N)}^{1/3}$.

Erdős, Sárközy and T. Sós [19, 21] asked whether there is a Sidon set which is also an asymptotic basis of order 3. Deshouillers and Plagne [54] proved in this direction that there is a Sidon set which is also an asymptotic basis of order 7, and Kiss [70] improved on this result by showing that there is a Sidon set which is also an asymptotic basis of order 5.

Answering a question of Sárközy, Ruzsa [82] showed that there is a set $\mathcal{A}\subset \{ 1,2,\ldots,n\}$ with $\vert \mathcal{A}\vert \geq \left (\frac{1} {2} + o(1)\right ){n}^{1/2}$ which is both additive and multiplicative Sidon set.

Improving on a result of Erdős, Sárközy and T. Sós [19, 20], Spencer and Tetali [91] showed that there exists an infinite Sidon set $\mathcal{A}$ such that any two consecutive elements s _i and s _i + 1 of the sum set $\mathcal{A} + \mathcal{A}$ satisfy $s_{i+1} - s_{i} < Cs_{i}^{1/3}\log s_{i}$ (for $i = 1,2,\ldots$) where C is an absolute constant.

As far as we know Problems 8–12 are still open.

In our original paper we mentioned the Erdős-Turán estimate (34) for the cardinality F(N, 1) of the largest Sidon set selected from $\{1,2,\ldots,N\}$. By (34) we have $F(N,1) = {N}^{1/2} + O({N}^{5/16})$. We remark that Babai and T. Sós [42] generalized the notion of Sidon set to groups and they studied the size of Sidon sets in groups. Among others, they proved that any finite group G has a Sidon subset of cardinality greater than c | G | ^1 ∕ 3. This seems to be quite far from being best possible, however, as far as we know it has not been sharpened yet.

A6. Difference-Sets

Some recent results and problems on the connection of sum sets and difference sets are discussed in the survey and problem papers by Martin and O’Bryant [75], Nathanson [76], Ruzsa [85] and Gyarmati, Hennecart and Ruzsa [61].

We do not know about any papers related to Problems 14–17.

A7. Generalizations

Horváth [67] proved partial results related to Problem 18; however, the problem is far from being settled.

On the other hand, we do not know about any papers related to Problems 19–24. In the case of the additive problems the reason of this is probably that the tools used in the special case of sums $a_{1} +\ldots +a_{k}$ fail when one tries to extend them to the general case $c_{1}a_{1} +\ldots +c_{k}a_{k}$. In the case of the multiplicative problems there does not seem to exist such a barrier, and one would expect that there is a better chance to achieve nontrivial results.

The problems of this type are getting quite popular.

Erdős, Sárközy and T. Sós [59] proved that for any $k \in \mathbb{N}$ and any k-colouring of $\mathbb{N}$, almost all the even numbers have a monochromatic representation in the form a + a′ with a≠a′. (This settled a conjecture of Roth.) In a recent paper Borbély [43] extended this result in various directions. (In another paper Erdős and Sárközy [56] also studied the multiplicative analogue of the problem in [59].)

Shkredov [90] proved both density results on the solvability of nonlinear equations of the type

$$\displaystyle{ f(a_{1},\ldots,a_{n}) = 0 }$$

(A7.1)

over $\mathbb{Z}/p\mathbb{Z}$ and the existence of monochromatic solutions of equations of this type.

Csikvári, Gyarmati and Sárközy [53] also studied both density and Ramsey-type problems involving equations of form (A7.1) over $\mathbb{Z}/m\mathbb{Z}$, $\mathbb{N}$ and $\mathbb{Q}$. Among others they extended Schur’s theorem [35] by proving that if $n,k \in \mathbb{N}$ and the prime p is large enough in terms of n and k, then for any k-colouring of $\mathbb{Z}/p\mathbb{Z}$ the Fermat equation

$$\displaystyle{{x}^{n} + {y}^{n} = {z}^{n}}$$

has a nontrivial monochromatic solution in $\mathbb{Z}/p\mathbb{Z}$. Moreover, they conjectured that for any k colouring of $\mathbb{N}$ the equation

$$\displaystyle{ a + b = cd,\quad a\neq b }$$

(A7.2)

has a monochromatic solution, and they proved partial results in this direction. Later Hindman [63] proved this conjecture in a more general form.

P. P. Pach [78] studied the following questions: is it true that if $k \in \mathbb{N}$, and $m \in \mathbb{N}$ is large enough, then the Eqs. (A7.2) and

$$\displaystyle{ ab + 1 = cd }$$

(A7.3)

have a “nontrivial” monochromatic solution in $\mathbb{Z}/m\mathbb{Z}$ for any k-colouring of it? He proved that in case of equation (A7.2) the answer is affirmative, while in case of equation (A7.3) one needs further assumptions on the prime factor structure of m to ensure the solvability.

Starting out from a problem of Pomerance and Schinzel, Sárközy asked the following question: is it true that for any r-colouring of the squarefree numbers greater than 1 the equation ab = c has a monochromatic solution? Pomerance and Schinzel [80] proved that the answer is affirmative for r = 2, and P. P. Pach [79] also proved this for r > 2.

2 A8. Probabilistic Methods. The Theorems of Erdős and Rényi

Dubickas [55] slightly sharpened Theorem 9 by showing that one can take $c_{1} {=\varepsilon }^{2}/10$ and $c_{2} = 2e+\varepsilon$ in the theorem for any $0 <\varepsilon < 1/2$.

Erdős and Rényi [13] also claimed in their paper that Theorem 10 can be extended from sums of two terms to sums of h terms (for fixed h), i.e., there is a similar theorem on B _h [λ] sets in place of B ₂[λ] sets. However, for h > 2 independence issues arise which are not at all easy to handle. This problem was cleared by Vu [95] who gave a complete and correct proof for the following theorem: for $h \in \mathbb{N}$ and h ≥ 2, and any $\varepsilon > 0$ there is a constant $g = g(\varepsilon )$ and a B _h [g] sequence $\mathcal{A}$ such that $A(x) \gg {x}^{1/h-\varepsilon }$, and, indeed, one can take $g_{h}(\varepsilon ) {\ll \varepsilon }^{-h+1}$. (See also the paper [52] of Cilleruelo, Kiss, Ruzsa and Vinuesa.)

We remark that the probabilistic approach is used in many of the papers mentioned in this Appendix.

At the end of Sect. 8 we mentioned a few papers to appear soon; these papers appear as Refs. [57, 91] and [58].

 ∗ 

We remark that the results described above induce many further problems. In a subsequent paper we will return to some of these problems and also present some related results.