Bárány et al.: Lagrange-like spectrum of perfect additive complements

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Full paper in Markdown. The arXiv record (https://arxiv.org/abs/2301.04365, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Balázs Bárány, Jin-Hui Fang, Csaba Sándor, "Lagrange-like spectrum of perfect additive complements," arXiv:2301.04365 (2023); published in Acta Arith. 212 (2024), 269--287, DOI 10.4064/aa230224-10-10. The copy read for this card is the arXiv version dated October 11, 2023.

Overview

The paper studies the possible normalized counting-function growth of perfect additive complements: infinite sets $A,B\subseteq\mathbb Z_{\ge 0}$ satisfying $r_{A,B}(n)=1$ for every $n\ge0$. Its object is

$$\mathfrak L=\left\{\limsup_{x\to\infty}\frac{A(x)B(x)}x:(A,B)\text{ are perfect additive complements}\right\},$$

expressed parametrically in the paper as $\limsup_k 2/(1+D_k)$. The mixed-radix classification (1.1) is quoted as Theorem A from [5, Theorem 1.1], rather than proved here. Likewise, Theorem B, quoted from [12, Lemma 2.1], gives

$$\limsup_{x\to\infty}\frac{A(x)B(x)}x=\limsup_{k\to\infty}\frac{2}{1+D_k},\qquad D_k=\frac1{m_k}-\frac1{m_km_{k-1}}+\cdots+\frac{(-1)^{k-1}}{m_k\cdots m_1};$$

see equations (1.1)–(1.2). Thus the paper’s new problem is the structure of the set of these limsup values, not the classification of perfect complements itself.

The principal result is Theorem 1.1. It proves that $\mathfrak L$ is closed; identifies its smallest accumulation point $\gamma_0\approx1.62688284$; and describes the spectrum below it as a strictly increasing explicit sequence

$$\frac32=\gamma_1<\frac85=\gamma_2<\frac{13}8=\gamma_3<\frac{109}{67}=\gamma_4<\cdots\to\gamma_0.$$

It also proves $[7/4,2]\subseteq\mathfrak L$, while $[12/7-\delta,2]\not\subseteq\mathfrak L$ for every $\delta>0$, and establishes that $[3/2,17/10]\cap\mathfrak L$ has Lebesgue measure zero. The exact lower endpoint $c_0=\inf\{c:[c,2]\subseteq\mathfrak L\}$ remains open between $12/7$ and $7/4$; see Problem 1.2. Dimension and continuity questions are posed in Problem 1.3.

The main reduction appears in Section 2. With $T_m(x)=(1-x)/m$, one has $D_k=T_{m_k}\circ\cdots\circ T_{m_1}(0)$. Defining

$$\mathcal L=\left\{\liminf_kT_{m_k}\circ\cdots\circ T_{m_1}(0):(m_i)\in\mathbb Z_{\ge2}^{\mathbb Z^+}\right\},$$

equation (2.3) gives $\mathfrak L=g(\mathcal L)$ for $g(x)=2/(1+x)$. The conjugate maps $\widehat G_m=g\circ T_m\circ g^{-1}$ yield the IFS representation (1.3). Theorems 2.1–2.5 are the corresponding statements for $\mathcal L$: closedness (Theorem 2.1), the interval $[0,1/7]\subseteq\mathcal L$ (Theorem 2.2), explicit gaps accumulating at $1/6$ (Theorem 2.3), measure zero on $[3/17,1/3]$ (Theorem 2.4), and the discrete portion above the first accumulation point (Theorem 2.5).

Section 3 proves Theorem 2.1 by concatenating increasingly long blocks from sequences realizing convergent spectral values; estimates (3.2)–(3.3) ensure that the resulting sequence has the desired liminf, as asserted in (3.1). Section 4 treats the interval and gap results. The attractor of $\{T_2,T_3,T_4\}$ is $[1/7,3/7]$ by (4.1); Lemmas 4.1–4.2 and the block construction following (4.2) prove Theorem 2.2. Lemma 4.3 restricts possible liminf values according to the symbols occurring infinitely often, while Lemma 4.4 and estimates (4.4)–(4.8) produce the gaps of Theorem 2.3.

Section 5 constructs words $M^{(1)}=2$, $M^{(2)}=3$, and $M^{(n)}=M^{(n-1)}M^{(n-2)}M^{(n-2)}$ in (5.5). If $\lambda_n=\operatorname{Fix}(T_{M^{(n)}})$, then $\lambda_n$ decreases to $\lambda_0=\Pi(M)=0.2293\ldots$. Proposition 5.3 proves $\lambda_n,\lambda_0\in\mathcal L$, using the cyclic-word ordering in Lemma 5.5; Proposition 5.6 excludes every interval $(\lambda_{n+1},\lambda_n)$ and all values above $\lambda_1$. Together these prove Theorem 2.5, and applying the decreasing map $g$ gives the sequence $\gamma_n=g(\lambda_n)$ in Theorem 1.1(2). Finally, Section 6 excludes the recurring block $(4,2)$ above $3/17$, embeds the remaining possibilities in a 55-map finite IFS, and applies the contraction-sum criterion of Lemma 2.1 to prove Theorem 2.4.

Relation to E1145

This source bears on Problem 1145.

For E1145, $1_A*1_B(n)$ is exactly the paper’s representation function $r_{A,B}(n)$. Consequently, every perfect complement in the paper would become a counterexample to E1145 if it also satisfied the balance condition on corresponding ordered elements.

The paper concerns pairs with $r_{A,B}=1$ and does not address E1145’s balance condition $a_n/b_n\to1$.