arXiv subjects
Publications and source records attributed to Laurence P. Wijaya.
We say that the set of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$ is super smooth if $y=\log^KN$ for a large fixed constant $K$. We show that the Roth's theorem on arithmetic progressions is true in super smooth numbers case. This extends the result of Harper where he showed the statement is true under a weaker hypothesis.
Recently there has been some progress in understanding the density of a subset of $[N]^2$ that avoids polynomial patterns. Kravitz, Kuca, and Leng showed that if $P\in\mathbb{Z}[z]$ satisfies certain conditions, then any set $A\subseteq[N]^2$ does not contain $(x,y),(x+P(z),y),(x,y+P(z))$, we must have \[ |A|\ll_P\frac{N^2}{(\log\log\log N)^c} \] for some small constant $c$. In this article, we show a similar result in $(\mathbb{F}_p)^2$ where we get a better bound on the density of a set $A\subseteq (\mathbb{F}_p)^2$ not containing $(x,y),(x+P(z),y),(x,y+P(z))$ with some conditions on $P\in \mathbb{F}_p[z]$.
We study the number of monochromatic solution to linear equation in $\{1,\dots,n\}$ when we color the set by at least three colors. We consider the $r$-commonness for $r\geq 3$ of linear equation with odd number of terms, and we also prove that any $2$-uncommon equation is $r$-uncommon for any $r\geq 3$.
Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $γ> 0$ and $x \ge 2$ we define $$H_γ(x)=\#\left\{m:~m=pq \le x, \ p 0$, uniformly over $γ$ with $$9/20+η\le γ\le 1/2 -η, $$ we have an asymptotic formula $$ H_γ(x)\sim C(γ)x^{1/2+γ}/ \log x, \qquad x \to \infty, $$ where $C(γ)> 0$ is an explicit constant depending only on $γ$. This extends the previous result of {É}.~Fouvry (2013), which has $12/25$ instead of $9/20$.
Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers.