Almost cubes and fourth powers in short intervals
Tsz Ho Chan
TL;DR
The paper investigates how short an interval $[x, x+x^{\theta_k}]$ must be to contain an integer of the form $n_1\cdots n_k$ with $n_i \asymp x^{1/k}$, extending the notion of almost squares to almost $k$th powers. The key innovation is a second moment method applied to Dirichlet polynomial constructions, enabling results that hold for all short intervals rather than almost all of them. Unconditionally, the author proves the existence of $n=m\cdot a_1a_2$ or $n=m\cdot a_1a_2a_3$ in intervals of lengths $x^{5/9+\varepsilon}$ and $x^{34/55+\varepsilon}$ under suitable density conditions on the $a_i$, and half- and full-density results for almost cubes with exponents $13/55+\varepsilon$. Conditional on Lindelöf, these exponents improve toward $x^{\varepsilon}$ for almost all $x$, highlighting a strong link between zero-free regions/bounds for $\zeta(s)$ and short-interval representations of almost $k$th powers. The methods combine Dirichlet polynomial mean-value theorems, a majorant principle, and Bourgain-type bounds for $\zeta(s)$, providing a versatile framework for studying multiplicative structures in short intervals.
Abstract
In this paper, we study how short an interval $[x, x + x^θ]$ contains an integer of the form $n_1 n_2 n_3$ and $m_1 m_2 m_3 m_4$ with $n_1 \approx n_2 \approx n_3$ and $m_1 \approx m_2 \approx m_3 \approx m_4$. The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals.