arXiv subjects

Neil Lyall

Publications and source records attributed to Neil Lyall.

23 records · Page 2Linked to original sources

arXiv2010

An optimal version of Sarkozy's theorem

Using Fourier analytic techniques, we prove that if $\VE>0$, $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,...,N\}$, then there must exist $t\in\N$ such that \[\frac{|A\cap (A+t^2)|}{N}>(\frac{|A|}{N})^2-\VE.\] This is a special case of results presented in Lyall and Magyar \cite{LM3} and we will follow those arguments closely. We hope that the exposition of this special case will serve to illuminate the key ideas contained in \cite{LM3}, where many of the analogous arguments are significantly more technical.

math.CA↗

arXiv2010

Simultaneous Polynomial Recurrence

Let $A\subseteq\{1,...,N\}$ and $P_1,...,P_\ell\in\Z[n]$ with $P_i(0)=0$ and $°P_i=k$ for every $1\leq i\leq\ell$. We show, using Fourier analytic techniques, that for every $\VE>0$, there necessarily exists $n\in\N$ such that \[\frac{|A\cap (A+P_i(n))|}{N}>(\frac{|A|}{N})^2-\VE\] holds simultaneously for $1\leq i\leq \ell$ (in other words all of the polynomial shifts of the set $A$ intersect $A$ "$\VE$-optimally"), as long as $N\geq N_1(\VE,P_1,...,P_\ell)$. The quantitative bounds obtained for $N_1$ are explicit but poor; we establish that $N_1$ may be taken to be a constant (depending only on $P_1,...,P_\ell$) times a tower of 2's of height $C_{k,\ell}^*+C\eps^{-2}$.

math.CA↗

arXiv2010

Arithmetic Structure in Sparse Difference Sets

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.

math.NT↗

arXiv2006

Strongly singular Radon transforms on the Heisenberg group and folding singularities

We prove sharp $L^2$ regularity results for classes of strongly singular Radon transfoms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.

math.CA↗

arXiv2006

Strongly singular integral operators associated to different quasi-norms on the Heisenberg group

In this article we study the behavior of strongly singular integrals associated to three different, albeit equivalent, quasi-norms on Heisenberg groups; these quasi-norms give rise to phase functions whose mixed Hessians may or may not drop rank along suitable varieties. In the particular case of the Koranyi norm we improve on previous arguments of the second author and obtain sharp $L^2$ estimates for the associated operators.

math.CA↗