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Source. Theorem 1, p. 12, of Jean-Marie De Koninck, Florian Luca and Igor E. Shparlinski, Powerful numbers in short intervals, Bull. Austral. Math. Soc. 71 (2005), 11--16, doi:10.1017/S0004972700037953. See the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof (pp. 12--14) was read for structure only. Nothing here is independently reviewed.

Statement

For an integer $\kappa>1$, an integer $m\ge1$ is $\kappa$-full when $p^\kappa\mid m$ for every prime $p$ dividing $m$ (p. 11); the $2$-full integers are the squarefull (powerful) ones.

Theorem 1 (p. 12). For every integer $\kappa\ge2$ there are infinitely many $N$ such that the open interval $(N^\kappa,(N+1)^\kappa)$ contains at least

$$M\ge\left(\left(\frac38+o(1)\right)\frac{\log N}{\log\log N}\right)^{1/3}$$

$\kappa$-full integers.

Here $\kappa$ is fixed, and the $o(1)$ term may depend on it (p. 14). The paper's closing remarks (pp. 15--16) say that running the argument with Liouville's theorem instead of Roth's gives a version explicit and uniform in $\kappa$, with $(3/8)^{1/3}$ replaced by $(3/8(\kappa-1))^{1/3}$; in particular, for infinitely many $N$ the interval $(N^{\kappa(N)},(N+1)^{\kappa(N)})$ then holds at least $(\log N)^{1/3+o(1)}$ $\kappa(N)$-full integers whenever $\kappa(N)=(\log N)^{o(1)}$, and arbitrarily many when $\kappa(N)=o((\log N)^{1/2})$. The remarks give no separate proof of these variants.

Proof pointer

Section 2 (pp. 12--14). Take $d_1<\cdots<d_{2\ell}$ the first $2\ell$ squarefree integers above $1$, their product $D$, and $\alpha_j=d_j^{-1/\kappa}$. A common denominator $q$, at least an explicit $R$ depending on $\kappa$ and $\ell$, approximates all the $\alpha_j$ simultaneously to within $q^{-1-1/2\ell}$; Dirichlet's simultaneous approximation theorem supplies such a $q$, and Roth's theorem applied to $\alpha_1$ bounds the least one. With $n=Dq$ the $2\ell$ distinct $\kappa$-full numbers $d_jD^\kappa r_j^\kappa$ all lie within $n^{\kappa-1}$ of $n^\kappa$, so one of $((n-1)^\kappa,n^\kappa)$ and $(n^\kappa,(n+1)^\kappa)$ holds at least $\ell$ of them. The size bound $n\le\exp((8(1+\delta)+o(1))\ell^3\log\ell)$ then converts $\ell$ into the stated count, $\delta>0$ being arbitrary.

Dependencies

Roth's theorem and Dirichlet's simultaneous approximation theorem, both cited from W. M. Schmidt, Diophantine approximation (Springer, 1980), Theorem 2A of Chapter 5 and Theorem 1A of Chapter 2.

Bears on

• Problem 942: the case $\kappa=2$ gives, for infinitely many $n$, at least $((3/8+o(1))\log n/\log\log n)^{1/3}$ powerful integers in $(n^2,(n+1)^2)$, hence in $[n^2,(n+1)^2)$. This is a lower bound for infinitely many $n$ only; it gives no upper bound valid for all $n$ and does not settle the problem. The paper does not mention the problem.