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Statement

Definition (Section 1, p. 45, after Hartman). Two infinite increasing sequences of integers $a_1<a_2<\cdots$ and $b_1<b_2<\cdots$ are far apart if, for every $A$, the inequality $\lvert a_i-b_j\rvert<A$ has only finitely many solutions.

Theorem (stated p. 45, proved pp. 45--47). Without using the axiom of choice, one can construct a set $\{a_t\}$ of real numbers of power $c$ (the power of the continuum) such that any two of the sequences $[a_t^n]$, $n=1,2,\ldots$, are far apart. The English summary (p. 48) states it as: "We show without using the axiom of choice that there exists a set $\{a_t\}$ of real numbers of power $c$ so that any two of the sequences $[a_t^n]$, $n=1,2,\ldots$ are far apart."

Context (p. 45). The paper credits Sierpiński with $c$ pairwise far-apart integer sequences, for instance $S_\alpha=\{2^{n+[n\alpha]}\}$, $n=1,2,\ldots$, for real $\alpha>0$, and Hartman with $\aleph_1$ reals $a_\alpha$, $1\le\alpha<\Omega_1$, whose sequences $[a_\alpha^n]$ are pairwise far apart. The theorem replaces Hartman's $\aleph_1$ by $c$ and needs no choice.

The construction (p. 45). With $u_n=2^{3^n}$ put

$$x_t=\sum_{n=1}^{\infty}\frac{1}{2^{[u_nt]}},\qquad \frac{9}{10}<t<1,$$

and $a_t=e^{x_t}$. The paper proves that no ratio $x_{t_1}/x_{t_2}$ with $9/10<t_1\ne t_2<1$ is a Liouville number, where (display (2)) a real $\alpha$ is a Liouville number if $\lvert\alpha-p/q\rvert<1/q^m$ is solvable in integers $p,q$ for every $m>0$.

Proof pointer

Hartman's remark (display (1), p. 45): if $[a^n]$ and $[b^n]$ are not far apart, then $\lvert\log a/\log b-p/q\rvert<1/b^p$ has infinitely many solutions in integers $p,q$, as one sees by taking logarithms in $\lvert a^q-b^p\rvert<A$. Since $\log a_{t_1}/\log a_{t_2}=x_{t_1}/x_{t_2}$, it suffices that no such ratio is a Liouville number.

That is shown with an auxiliary lemma (p. 46), which the paper calls known and proves for completeness: if a real $\alpha$ has reduced approximations $a_n/b_n$ with $b_1<b_2<\cdots$, $\lvert\alpha-a_n/b_n\rvert<1/(2b_n^2)$ and $b_{n+1}<b_n^{c_1}$ for an absolute constant $c_1$, then $\alpha$ is not a Liouville number. For $9/10<t_1<t_2<1$ the reduced fractions $a_n'/b_n'$ equal to the ratio of the $n$-th partial sums of $x_{t_2}$ and $x_{t_1}$ approximate $x_{t_2}/x_{t_1}$ well enough (displays (4), (5)), and after ordering, the denominators grow at most polynomially from one to the next (display (6), proved by (7)--(10) on pp. 46--47); the lemma then applies, and the case $x_{t_1}/x_{t_2}$ follows because $1/\alpha$ is a Liouville number whenever $\alpha$ is.

Question raised (p. 47)

In connection with the proof the paper asks for a field of real numbers of power $c$ containing no irrational Liouville number, and adds that it could not even prove that a ring of real numbers of power $c$ without irrational Liouville numbers exists. The English summary (p. 48) puts it as: "Does there exist a field of real numbers of power $c$ no element of which is a Liouville number? I could not decide this question." By definition (2) every rational is a Liouville number, so the Hebrew text's qualifier irrational is the reading under which the question is not trivially answered no.

Read depth. Claims checked: the definition, the theorem, the construction, the lemma and the question were read clause by clause on the page images of pp. 45--48; the proof was followed in outline and its estimates were not re-derived.

Source. P. Erdős, Some remarks on number theory (in Hebrew), Riveon Lematematika 9 (1955), 45--48; the edition read is named on the source card.

Dependencies

Hartman's remark (1) and the lemma of p. 46, both inside the paper.

Bears on

No Erdős problem in this corpus is borne on by this result.