Wang: A Proposed Solution to Erdős Problem 486

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Shouqiao Wang, "A Proposed Solution to Erdős Problem 486," preprint, 2026.

Overview

Wang studies the delayed multi-residue sieve obtained from an arbitrary set of moduli $A\subseteq\mathbb N$ and subsets $X_n\subseteq\mathbb Z/n\mathbb Z$:

$$B(A,X)=\{m\in\mathbb N:m\bmod n\notin X_n\text{ for every }n\in A\text{ with }n<m\}.$$

This is equation (1.1) in Section 1. The question is whether $L_B(x)=(\log x)^{-1}\sum_{m<x,\,m\in B}m^{-1}$ must converge. Theorem 1.1 gives a negative answer: there are fixed infinite $A$ and fixed sets $X_n$ for which

$$\liminf_{x\to\infty}L_B(x)\le 177/200<49/50\le\limsup_{x\to\infty}L_B(x).$$

The result is a counterexample to Problem 486. The manuscript attributes the original formulation to Erdős [7, p. 48] and [8, pp. 235--236].

The proof combines finite probabilistic deletion blocks with a deterministic gliding-hump construction. Lemma 2.1 in Section 2 is the recovery mechanism: for any finite family $\mathcal F=\{(q,Y_q)\}$, if $U_{\mathcal F}\subset\widehat{\mathbb Z}$ is the union of the associated residue cylinders, then

$$\sum_{m<x,\,m\in B_{\mathcal F}}\frac1m =(1-\mu(U_{\mathcal F}))\log x+O_{\mathcal F}(1).$$

The proof reduces the eventual survivor set to a union of residue classes modulo the least common multiple of the finitely many moduli. Remark 1.2 shows that replacing the strict activation condition $q<m$ by the inclusive condition $q\le m$ changes the survivor set by a primitive set and hence by logarithmic weight $o(\log x)$, using Behrend [3, pp. 42--44].

Section 3 constructs one block at the dyadic scale $Q=2^j$. Lemma 3.1 associates distinct moduli $q_S$ to central subsets $S\subseteq\{1,\ldots,k\}$, where $k=2\lfloor\sqrt j/8\rfloor$, so that

$$19Q/20\le q_S\le21Q/20, \qquad p_i\mid q_S\Longleftrightarrow i\in S, \tag{3.1}$$

and every point of $J=[11Q/10,19Q/10]\cap\mathbb Z$ lies in $(q_S,2q_S]$. Independent Bernoulli labels modulo the auxiliary primes define $K(m)$ and select endpoints $E=\{m\in J:K(m)\in\mathcal S_k\}$, assigning $q_m=q_{K(m)}$. Lemma 3.2, by McDiarmid's bounded-differences inequality, gives

$$\mathbb P(|E|<|J|/2)\le \exp(-4^k/(8k)).$$

Lemma 3.3 proves that the completed periodic footprint $U=\bigcup_{m\in E}[m]_{q_m}$ satisfies

$$\mathbb P(\mu(U)>e^{-k/100})\le3e^{-k/100}.$$

Its main ingredients are the candidate characterization (3.2), the collision estimate (3.3), the one-candidate probability (3.4), Hoeffding's tail bound (3.5), and the entropy and candidate-count estimates (3.6) and (3.7). These are probabilistic existence arguments; no explicit labels or blocks are computed.

Lemma 3.4 extracts a deterministic block: for every sufficiently large $j$, there are $E_j\subset[11Q/10,19Q/10]$, assigned moduli $q_{j,m}<m$, and a cylinder union $U_j$ such that

$$|E_j|\ge3Q/8,\qquad 19Q/20\le q_{j,m}\le21Q/20, \qquad \mu(U_j)\le\eta_j=e^{-k_j/100}, \tag{3.8}$$

with $\sum_j\eta_j<\infty$. Thus the active classes delete a fixed amount of local harmonic mass while their eventual periodic union has summably small Haar measure.

Section 4 assembles blocks in epochs $I_t=\{a_t,\ldots,2a_t\}$. The tail bounds in (4.1) keep the accumulated periodic footprint below $\epsilon=1/100$. Lemma 2.1 permits the next epoch to be delayed until the finite past has recovered: at $x_t=2^{a_t-1}$, equation (4.2) gives a finite-past average at least $49/50$. Every modulus from epoch $t$ or later exceeds $x_t$, so it is inactive below $x_t$, and the scale-separation inequality (4.3) keeps later scales from adding classes to earlier moduli; this gives the limsup bound (4.4). Conversely, all endpoints in epoch $t$ have been deleted by $y_t=2^{2a_t+1}$. Their harmonic contribution is bounded below scale by scale, yielding (4.5) and the liminf bound (4.6). These two subsequences prove Theorem 1.1.

The construction lies outside the known positive regimes. Section 1 cites the Davenport--Erdős theorem for the zero-residue case [5, pp. 147--151], its later elementary proof [6, pp. 19--24], Besicovitch's failure of natural density [2, pp. 336--341], and Araújo's summable multi-residue result [1, Theorem 3.25]. Remark 4.1 proves for Wang's system that every installed scale contributes at least $5/14$ to $\sum_{q\in A}|X_q|/q$, so that series diverges.

Relation to E25

This source bears on Problem 25.

Enumerate Wang's modulus set increasingly as $n_1<n_2<\cdots$. E25 corresponds to imposing

$$X_{n_i}=\{a_i\bmod n_i\}$$

for every $i$. Apart from activation at equality, its uncovered set is then Wang's $B(A,X)$: E25 tests the congruence when $n=n_i$, whereas equation (1.1) activates a modulus only when $n_i<n$. Remark 1.2 proves that these strict and inclusive conventions have the same logarithmic-density behavior, since their difference has harmonic sum $o(\log x)$. Activation convention is therefore not the obstruction to transferring the counterexample.

The paper's blocks are not singleton blocks. In Lemma 3.4, $q_{j,m}=q_{K(m)}$, so all endpoints with the same label set $K(m)=S$ use the same modulus and become distinct elements of the multi-residue set $X_{q_S}$. There are at most $|\mathcal S_k|\le2^k=\exp(O(\sqrt j))$ available moduli at scale $j$, while $|E_j|\ge3\cdot2^j/8$. Retaining at most one endpoint per modulus would therefore leave only $\exp(O(\sqrt j))$ endpoints of size $\asymp2^j$, losing the fixed harmonic deletion required in (4.5). Remark 4.1 also explicitly counts the many residue classes through $|X_q|$. No argument is given for replacing repeated uses of $q_S$ by distinct moduli while preserving the small-footprint estimate of Lemma 3.3.

Consequently, Theorem 1.1 is a counterexample only for the multi-residue generalization.