Finding almost squares IV

Abstract.

In this paper, we continue the study of almost squares; these are integers n representable as n = a · b for some $a, b \in [1,3\sqrt{n}]$. We show that almost all (in the measure–theoretic sense) short intervals [x, x + (log x)¹²] contain at least one almost square, and we consider related questions. Moreover, a result of Erdős shows that the exponent 12 cannot be smaller than 0.086 ....