Fang–Sándor: On sets with sum and difference structure

../

---

Full paper in Markdown.

Jin-Hui Fang, Csaba Sándor, "On sets with sum and difference structure," arXiv:2205.06553 (2022).

Overview

Fang and Sándor study the representation functions

$$r_{A,B}(n)=\#\{(a,b)\in A\times B:a+b=n\},\qquad d_{A,B}(n)=\#\{(a,b)\in A\times B:a-b=n\}$$

for nonempty sets of nonnegative integers. Their principal question is the rigid extremal case in which every nonnegative integer has exactly one representation as a sum. The retained folder-name PDF is arXiv:2205.06553v1 (13 May 2022), 7 pages; the locators below refer to its numbered results, equations, and sections. The arXiv record (https://arxiv.org/abs/2205.06553, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The main classification, Theorem 1.1, says that $r_{A,B}(n)=1$ for every $n\geq0$ if and only if, up to interchanging $A$ and $B$, the two sets split the alternating digit positions of a mixed-radix expansion. More explicitly, for integers $m_j\geq2$ and $M_j=m_1\cdots m_j$ ($M_0=1$),

$$A=\left\{\sum_{i\geq0}\varepsilon_{2i}M_{2i}:0\leq\varepsilon_{2i}<m_{2i+1}\right\},\qquad B=\left\{\sum_{i\geq0}\varepsilon_{2i+1}M_{2i+1}:0\leq\varepsilon_{2i+1}<m_{2i+2}\right\},$$

with only finitely many nonzero digits; this is formula (1.2). Sufficiency follows from uniqueness of mixed-radix expansion. Necessity is proved in §2 by repeatedly applying Lemma 2.1, which extracts an initial digit block: if $1\in C$ and $r_{C,D}(n)=1$ for every $n\geq0$, then for some $m\geq2$ one has $C=\{0,\ldots,m-1\}+mE$ and $D=mF$, with $r_{E,F}\equiv1$. The unnumbered Proposition in the proof of Theorem 1.1 iterates this decomposition; its finite-stage sets are displayed in (2.1), and passage to arbitrarily many stages yields (1.2).

Theorem 1.3 proves that these exact complements also have unique differences: $d_{A,B}(n)=1$ for every integer $n$. The proof establishes the explicit interval identity $A_k-B_k=I_k$ in (2.2); since the number of pairs equals the interval length, every element of $I_k$ has one difference representation, and the increasing finite stages exhaust $A$ and $B$. The converse observation immediately following Theorem 1.3 is also useful: if additive complements satisfy $d_{A,B}(n)\leq1$ for every integer $n$, then their sum representations must all be unique, because two different decompositions $a+b=a'+b'$ produce the repeated difference $a-b'=a'-b$.

Theorem 1.5 quantifies the counting functions of exact complements:

$$\liminf_{x\to\infty}\frac{A(x)B(x)}x=1, \qquad \frac32\leq\limsup_{x\to\infty}\frac{A(x)B(x)}x\leq2,$$

and asserts that both endpoint constants are sharp. The liminf is attained along $x_k=M_{2k}-1$, where $A(x_k)B(x_k)=x_k+1$. The limsup calculation uses Lemma 2.2, explicitly imported from [3, Lemma 2.1], and its formula (2.3) in terms of the alternating quantities $D_k$. Remark 1.6 consequently rules out simultaneous global uniqueness and the asymptotic relation $A(x)B(x)\sim x$.

Theorem 1.7 gives a contrasting existence result: there are additive complements with $A(x)B(x)=(1+o(1))x$ for which, for every prescribed integer $c\geq0$, the equality $d_{A,B}(n)=c$ occurs for infinitely many positive $n$. Its proof starts from the complements supplied by the cited result [2, Theorem 2] (stated in the introduction as Theorem B), adjoins sparse finite configurations $C_n,D_n$ near $T_n=\max\{a_{n^4},b_{n^4}\}$, and controls the perturbation by

$$A_1(x)\leq A(x)\leq A_1(x)+\sqrt{A_1(x)},\qquad B_1(x)\leq B(x)\leq B_1(x)+\sqrt{B_1(x)}.$$

Danzer’s Theorem A and Theorem B are cited background rather than results proved here. Problems 1.2 and 1.4 are explicitly open questions posed by the authors: they ask whether complements not of the mixed-radix form must have, respectively, at least two sum representations or at least two difference representations for infinitely many integers. Neither question is resolved in the paper.

Relation to E1145

This source bears on Problem 1145.

In E1145 notation, the paper’s $r_{A,B}(n)$ is exactly the convolution $(1_A*1_B)(n)$. Its difference function is $(1_A*1_{-B})(n)$. Passing from nonnegative sets to positive sets by $A^+=A+1$ and $B^+=B+1$ gives

$$(1_{A^+}*1_{B^+})(n)=r_{A,B}(n-2),$$

so the mixed-radix systems of Theorem 1.1 furnish positive sets with exactly one representation of every $n\geq2$. Thus additive-complement status alone cannot imply E1145’s conclusion.

The simplest instance makes clear why this is not a counterexample to E1145. Taking every $m_j=2$ in (1.2), $A$ consists of integers whose binary digits occur only in even positions and $B=2A$. If $A^+=A+1$ and $B^+=B+1$, their corresponding positive enumerations satisfy $a_n/b_n\to1/2$ (or $2$ after interchange), not $1$.

Theorem 1.1 is usable as a rigidity lemma only under the stronger hypothesis of unique representation from the bottom: any proposed globally unique counterexample must be an alternating mixed-radix construction. E1145, however, assumes only eventual coverage and asks whether the representation function can be bounded by an arbitrary constant; the classification does not cover eventual uniqueness, finite exceptional ranges, or bounds $r_{A,B}\leq M$ with $M>1$.

Theorem 1.5 offers a possible obstruction in a strengthened argument: global uniqueness forces $\limsup A(x)B(x)/x\geq3/2$, so any independent consequence of E1145’s balance condition forcing $A(x)B(x)/x\to1$ would contradict uniqueness. The paper proves no such consequence and never relates $a_n/b_n\to1$ to its mixed-radix parameters or counting-function estimates.

Theorem 1.7 is mainly a warning about scope: near-minimal counting product $A(x)B(x)\sim x$ permits highly flexible difference multiplicities. It supplies neither $a_n/b_n\to1$ nor an upper bound for the sum representation function. Likewise, Problems 1.2 and 1.4 seek only multiplicity at least two infinitely often and remain conjectural; even affirmative answers would be far weaker than E1145’s required $\limsup_n(1_A*1_B)(n)=\infty$. Consequently, this paper provides structural tests and extremal examples relevant to counterexample design, but no direct progress from the balance hypothesis to unbounded additive multiplicity.