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Statement

Setting (p. 107). $s_1<s_2<\cdots$ are the squarefree numbers, and $g_t(x)$ is the number of $s_i<x$ with $s_{i+1}-s_i=t$.

Lemma 1 (p. 107), cited as known from Mirsky (footnote 3): for fixed $t$, as $x\to\infty$, $g_t(x)=\beta_tx+o(x)$; that is, the density $\beta_t$ of the $s_i$ with $s_{i+1}-s_i=t$ exists.

The moment asymptotic (23) (p. 107). Erdős says that, on hearing of Roth's bound (22), he thought of trying to prove for every $\alpha$ that

$$\sum_{s_{i+1}\le x}(s_{i+1}-s_i)^\alpha=C_\alpha x+o(x).\qquad(23)$$

He says the proof of (23) seems very difficult, notes that it would imply $s_{i+1}-s_i=o(s_i^\varepsilon)$, and states that he can prove (23) only for $\alpha<A$, where $A$ is a certain constant between 2 and 3. The paper sketches only the case $\alpha=2$.

The case α = 2 (p. 109). The series $\sum_{t\ge1}t^2\beta_t$ converges, and

$$\sum_{s_{i+1}\le x}(s_{i+1}-s_i)^2=x\sum_{t=1}^{\infty}t^2\beta_t+o(x).$$

The paper says this "proves (23) for $\alpha=2$". It gives no argument for other exponents, and the constant $A$ is not identified.

Source. P. Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109, doi:10.5486/pmd.1951.2.2.04: Lemma 1 and (23) on p. 107, the sketch on pp. 108--109. Lemma 1 is cited there from L. Mirsky, Arithmetical pattern problems relating to divisibility by $r$-th powers, Proc. London Math. Soc. 50 (1949), 497--508, Theorem 4, p. 507. The edition read is identified on the source card.

Read depth. Claims checked: the statements of Lemma 1, (23) and the case $\alpha=2$ were read clause by clause on the printed pages, and the sketch was read through but not checked step by step. Lemma 1 was not checked against Mirsky's paper. Nothing here is independently reviewed.

Proof pointer

Pages 108--109. From Lemma 2, grouping the gaps into dyadic ranges $(2^{r+j},2^{r+j+1}]$, for every $\varepsilon>0$ there is an $r$ with $\sum_{s_{i+1}\le x,\ s_{i+1}-s_i>2^r}(s_{i+1}-s_i)^2<\varepsilon x$ (28). From Lemma 1, the sum over gaps at most $2^r$ is $x\sum_{t\le2^r}t^2\beta_t+o(x)$ (29). Together these give a bound $O(x)$ for the full sum, the convergence of $\sum t^2\beta_t$, and the asymptotic.

Dependencies

Lemma 2 of the same paper, and Lemma 1 from Mirsky's paper cited above.

Bears on

• Problem 145: the case $\alpha=2$ shows that the limit the problem asks about exists for $\alpha=2$, with value $\sum t^2\beta_t$. The paper states, without proof, that it can reach every $\alpha<A$ with $A$ between 2 and 3. The range $0\le\alpha\le2$ that the claim page Erdős 1951 records is the range later papers credit to this one; the paper prints only the case $\alpha=2$.
• Problem 489: the case $A=\{p^2:p\text{ prime}\}$, whose $B$ is the squarefree numbers and which meets the problem's hypothesis $|A\cap[1,x]|=o(x^{1/2})$; for it the mean squared gap tends to the finite limit $\sum t^2\beta_t$. The paper's sum runs over $s_{i+1}\le x$ where the problem's runs over $b_i<x$. It says nothing about any other $A$ beyond the remarks on p. 109.