Fang–Sándor: On function $SX$ of additive complements

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Full paper in Markdown. The arXiv record (https://arxiv.org/abs/2210.09680, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Jin-Hui Fang, Csaba Sándor, "On function $SX$ of additive complements," arXiv:2210.09680 (2022).

Overview

Fang and Sándor study additive complements $A,B\subseteq\mathbb N_0$ through their counting functions $A(x),B(x)$ and the Erdős–Freud quantity

$$SX(A,B)=\limsup_{x\to\infty}\frac{\max\{A(x),B(x)\}}{\sqrt{x}}.$$

The starting point is Narkiewicz’s cited result, reproduced as Theorem A in §1: if $A(x)B(x)=(1+o(1))x$, then $\log\min\{A(x),B(x)\}/\log x\to0$. This is background rather than a result proved in the paper.

Theorem 1.1 (§1, proved in §2) gives a quantitative extension. If

$$\limsup_{x\to\infty}\frac{A(x)B(x)}x=1+\delta, \qquad 0\leq\delta<C_0,$$

where

$$C_0=\tfrac12\bigl(-3-\sqrt2+\sqrt{3+12\sqrt2}\bigr)=0.027315\ldots,$$

then

$$\limsup_{x\to\infty}\frac{\log\min\{A(x),B(x)\}}{\log x}\leq f(\delta),$$

with

$$f(\delta)=\log_2\!\frac{(2\delta+2)^2}{3-2\delta+\sqrt{(3-2\delta)^2-(2\delta+2)^3}}.$$

The remark after Theorem 1.1 records $f(0)=0$, $f(C_0)=1/2$, and monotonicity on $[0,C_0]$. Consequently, Corollaries 1.2 and 1.3 assert that, under the strict inequality $\limsup A(x)B(x)/x<1+C_0$,

$$\frac{\min\{A(x),B(x)\}}{\sqrt x}\longrightarrow0, \qquad \frac{\max\{A(x),B(x)\}}{\sqrt x}\longrightarrow\infty.$$

The proof counts pairs in $A\cap[0,x]$ and $B\cap[0,x]$, separating those with sum at most $x$ from pairs whose two coordinates exceed $x/2$. This yields a quadratic restriction on the dilation ratios $A(x)/A(x/2)$ and $B(x)/B(x/2)$; iteration along dyadic scales produces the exponent $f(\delta)$. The paper itself (arXiv:2210.09680v1, pp. 3–4) prints three slips in this proof: the displayed polynomial $p_\delta$ has a sign incompatible with its subsequently displayed roots, both dilation alternatives carry an extraneous exponent $x$, and the final display has $\min\{A(x),B(x)\}/\log x$ where the theorem requires logarithms in the numerator. The theorem statement and the root formulas nevertheless identify the intended estimate.

Using the elementary covering inequality $A(x)B(x)\geq x-O(1)$, Theorem 1.4 (§1, proved in §2) establishes the unconditional bound

$$SX(A,B)\geq\sqrt{1+C_0}=1.013565\ldots$$

for every pair of additive complements. The proof divides according as $\limsup A(x)B(x)/x$ is below $1+C_0$, when Corollary 1.3 makes $SX$ infinite, or at least $1+C_0$, when $\max\{A(x),B(x)\}\geq\sqrt{A(x)B(x)}$ gives the stated constant.

The second part concerns perfect additive complements, meaning that every nonnegative integer has exactly one representation $a+b$. The paper recalls from reference [3], rather than reproving, their mixed-radix structure (1.1). Writing $P_j=m_1\cdots m_j$ and $P_0=1$, the two sets use respectively the even and odd digit positions:

$$A=\left\{\sum_{i\geq0}\epsilon_{2i}P_{2i}:0\leq\epsilon_{2i}<m_{2i+1}\right\},\qquad B=\left\{\sum_{i\geq1}\epsilon_{2i-1}P_{2i-1}:0\leq\epsilon_{2i-1}<m_{2i}\right\},$$

up to interchanging $A$ and $B$, where every $m_j\geq2$.

Theorem 1.5 gives an exact formula for $SX$ for these systems. If

$$X_s^A=\sum_{i=1}^s(m_{2i-1}-1)P_{2i-2},\qquad X_s^B=\sum_{i=1}^s(m_{2i}-1)P_{2i-1},$$

then

$$SX(A,B)=\limsup_{s\to\infty}\max\left\{ \frac{\prod_{i=1}^s m_{2i-1}}{\sqrt{X_s^A}}, \frac{\prod_{i=1}^s m_{2i}}{\sqrt{X_s^B}} \right\}.$$

The proof first evaluates the counting functions at the two full-digit endpoints. It then shows that the ratios $A(y)/\sqrt y$ and, analogously, $B(z)/\sqrt z$ can be increased by filling the first incomplete relevant digit; the requisite digit expansions are (2.1) and (2.2). Thus the endpoint values control the full limsup. The paper's display following (2.2) (p. 5) writes $A(z)$ although the symmetry of the argument calls for $B(z)$.

Theorem 1.6 determines the sharp infimum over perfect additive complements:

$$\inf SX(A,B)=\sqrt[4]{4.5}.$$

The upper approximation takes $m_n=2$ for $n\geq3$ and chooses initial integers $m_1^{(k)},m_2^{(k)}$ with $m_2^{(k)}/m_1^{(k)}\to\sqrt2$. For the lower bound, the formula of Theorem 1.5 is rewritten in terms of reciprocal radix products. The proof separates the cases where infinitely many even radices are at least $3$, where eventually all even radices are $2$ but infinitely many odd radices are at least $3$, and where all radices from $m_3$ onward equal $2$; the last case reduces to the two quantities $\frac13(m_2/m_1)$ and $\frac23(m_1/m_2)$ and the inequality $\min\{u,v\}\leq\sqrt{uv}$. The remark after Theorem 1.6 observes that $SX$ is unbounded above among perfect complements, for example when $m_{2i-1}=2$ and $m_{2i}=3$.

The paper does not claim that its constants are optimal for arbitrary additive complements. Problem 1.7 asks whether positive lower counting exponents can occur with $\limsup A(x)B(x)/x$ arbitrarily close to $1$, and Problem 1.8 asks whether the perfect-complement lower bound $\sqrt[4]{4.5}$ holds for all additive complements. These are explicitly posed problems, not proved assertions.

Relation to E1145

This source bears on Problem 1145.

For E1145, write

$$A^{\#}(x)=|A\cap[1,x]|, \qquad B^{\#}(x)=|B\cap[1,x]|, \qquad r_{A,B}(n)=(1_A*1_B)(n).$$

The paper’s additive-complement hypothesis is exactly the eventual covering condition $r_{A,B}(n)\geq1$ in E1145, apart from its harmless convention of allowing $0$.

The balance condition $a_n/b_n\to1$ is a condition on inverse counting functions, not literally the assertion $A^{\#}(x)/B^{\#}(x)\to1$. Precisely, for every $\eta>0$ and all sufficiently large $x$ one obtains, up to finitely many initial elements,

$$A^{\#}(x)\leq B^{\#}((1+\eta)x)+O(1), \qquad B^{\#}(x)\leq A^{\#}((1+\eta)x)+O(1).$$

Large local gaps or clusters prevent replacing these dilated comparisons by same-point asymptotic equality without an additional regularity hypothesis.

There is nevertheless a concrete necessary condition for a counterexample to E1145. If $\limsup_n r_{A,B}(n)<\infty$, choose a uniform bound $M$. Counting all pairs with $a,b\leq x$ gives

$$A^{\#}(x)B^{\#}(x) \leq\sum_{n\leq2x}r_{A,B}(n) \leq2Mx+O(1),$$

whereas eventual covering gives $A^{\#}(x)B^{\#}(x)\geq x-O(1)$. Combining the upper estimate with the dilated comparisons implied by $a_n/b_n\to1$ shows individually

$$A^{\#}(x),B^{\#}(x)=O(\sqrt x),$$

and then the covering lower bound gives $A^{\#}(x),B^{\#}(x)=\Omega(\sqrt x)$. Thus any counterexample to E1145 would have both counting functions of square-root order.

Corollary 1.3 can then be used as an exclusion criterion: such a balanced bounded-representation pair cannot satisfy

$$\limsup_{x\to\infty}\frac{A^{\#}(x)B^{\#}(x)}x<1+C_0,$$

because the corollary would force $\max\{A^{\#}(x),B^{\#}(x)\}/\sqrt x\to\infty$, contradicting the preceding $O(\sqrt x)$ bound. Hence every hypothetical counterexample must obey

$$\limsup_{x\to\infty}\frac{A^{\#}(x)B^{\#}(x)}x\geq1+C_0.$$

This is a genuine restriction, but it is far from forcing $r_{A,B}(n)$ to be unbounded.

Perfect complements are the extremal obstruction motivating E1145: they have $r_{A,B}(n)=1$. A perfect pair on $\mathbb N_0$ may be shifted to positive sets $A+1,B+1$, producing exactly one representation of every integer $n\geq2$; the shift does not affect asymptotic sequence ratios or $SX$. Consequently, if any mixed-radix pair (1.1) also satisfied $a_n/b_n\to1$, it would furnish a counterexample to E1145. Theorem 1.5 provides an exact counting-function formula with which such candidates can be tested, and Theorem 1.6 says that every exact perfect candidate has

$$SX(A,B)\geq\sqrt[4]{4.5}.$$

However, neither theorem analyzes the enumerated-term ratio $a_n/b_n$, and the paper does not prove that balanced perfect complements exist or that they are impossible.

More generally, the recalled classification (1.1) applies only to exact unique representation of every nonnegative integer. It does not classify pairs having merely bounded multiplicity, or even pairs that are uniquely representing only for all sufficiently large integers. The paper’s $SX$ bounds control the size of counting functions, not collisions among sums. Accordingly, the paper supplies useful density obstructions and a structured family of potential extremal examples, but it does not establish the conclusion $\limsup_n r_{A,B}(n)=\infty$ under E1145’s balance hypothesis.