arXiv subjects

Khalid Younis

Publications and source records attributed to Khalid Younis.

3 recordsLinked to original sources

arXiv2026

Sets whose differences avoid a bracket quadratic

Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).

math.NT↗

arXiv2024

Asymptotics for smooth numbers in short intervals

A number is said to be $y$-smooth if all of its prime factors are less than or equal to $y.$ For all $17/30<θ\leq 1,$ we show that the density of $y$-smooth numbers in the short interval $[x,x+x^θ]$ is asymptotically equal to the density of $y$-smooth numbers in the long interval $[1,x],$ for all $y \geq \exp((\log x)^{2/3+\varepsilon}).$ Assuming the Riemann Hypothesis, we also prove that for all $1/2<θ\leq 1$ there exists a large constant $K$ such that the expected asymptotic result holds for $y\geq (\log x)^{K}.$ Our approach is to count smooth numbers using a Perron integral, shift this to a particular contour left of the saddle point, and employ a zero-density estimate of the Riemann zeta function.

math.NT↗

arXiv2019

Lower bounds in the polynomial Szemerédi theorem

We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's construction for sets lacking a square difference. We also extend Ruzsa's construction to sets lacking polynomial differences for a wide class of univariate polynomials. Finally, we turn to multivariate differences, constructing a set of order $N^{1/2}$ lacking a difference equal to a sum of two squares. This is in contrast to the analogous problem of sets lacking a difference equal to a prime minus one, where the current record is of order $N^{o(1)}.$

math.NT↗