Finding almost squares

@article{Chan2006FindingAS,
  title={Finding almost squares},
  author={Tsz Ho Chan},
  journal={Acta Arithmetica},
  year={2006},
  volume={121},
  pages={221-232}
}
  • T. Chan
  • Published 9 February 2005
  • Mathematics
  • Acta Arithmetica
We study short intervals which contain an ``almost square'', an integer $n$ that can be factored as $n = ab$ with $a$, $b$ close to $\sqrt{n}$. This is related to the problem on distribution of $n^2 \alpha \pmod 1$ and the problem on gaps between sums of two squares. 
Finding Almost Squares III
An almost square of type 2 is an integer $n$ that can be factored in two different ways as $n = a_1 b_1 = a_2 b_2$ with $a_1$, $a_2$, $b_1$, $b_2 \approx \sqrt{n}$. In this paper, we shall improve
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
Finding Almost Squares V
TLDR
The study of almost squares of type 2 in short intervals is continued and the 1/2 upper bound is improved and connections with almost square of type 1 are drawn.
Finding almost squares VI
  • T. Chan
  • Mathematics
    International Journal of Number Theory
  • 2018
In this paper, we continue the study of almost squares and extend the result of the author’s fourth paper in the series to almost squares with closer factors. We prove that almost all intervals
Distance between arithmetic progressions and perfect squares
In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in
J an 2 01 8 Distance between arithmetic progressions and perfect squares Tsz
In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in
NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS
We study systems of $n$ points in the Euclidean space of dimension $d\geqslant 1$ interacting via a Riesz kernel $|x|^{-s}$ and confined by an external potential, in the regime where $d-2\leqslant

References

SHOWING 1-10 OF 10 REFERENCES
Some problems of diophantine approximation
Let 0 be an irrational number, and a any number between 0 and 1 (0 included). Then it is well known that it is possible to find a sequence of positive integers w<i, n2, n3, ... such that (nr0) —>a as
ON THE DISTRIBUTION OF THE SEQUENCE n2θ (mod 1)
Small values of n 2 α (mod 1), Invent
  • Math
  • 1995
On the distribution of the sequence n 2 θ (mod 1), Canad
  • J. Math
  • 1987
An asymptotic inequality in the theory of numbers, Vestnik Leningrad
  • Univ
  • 1960
An asymptotic inequality in the theory of numbers
  • Vestnik Leningrad. Univ
  • 1960
On the distribution of the sequence n 2 θ (mod 1), Quart
  • J. Math. (Oxford)
  • 1948
American Institute of Mathematics 360 Portage Avenue Palo Alto, CA 94306 USA thchan@aimath
  • American Institute of Mathematics 360 Portage Avenue Palo Alto, CA 94306 USA thchan@aimath
Small values of n2α
  • Invent. Math
  • 1995