arXiv subjects
Publications and source records attributed to Akos Magyar.
Let A be a subset of positive relative upper density of P^d, the d-tuples of primes. We prove that A contains an affine copy of any finite set of lattice points E, as long as E is in general position in the sense that it has at most one point on every coordinate hyperplane.
We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial Szemeredi Theorem asserting that the difference set of any subset of the integers of positive upper density necessarily contains a perfect square) and then applying a simple lifting argument.
Let $P\in\Z[n]$ with $P(0)=0$ and $\VE>0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,\...,N\}$, then there must exist $n\in\N$ such that \[\frac{|A\cap (A+P(n))|}{N}>(\frac{|A|}{N})^2-\VE.\] In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\N$, then the set of \emph{$\VE$-optimal return times} \[R(A,P,\VE)=\{n\in \N \,:\,\D(A\cap(A+P(n)))>\D(A)^2-\VE\}\] is syndetic for every $\VE>0$. Moreover, we show that $R(A,P,\VE)$ is \emph{dense} in every sufficiently long interval, in the sense that there exists an $L=L(\VE,P,A)$ such that \[|R(A,P,\VE)\cap I| \geq c(\VE,P)|I|\] for all intervals $I$ of natural numbers with $|I|\geq L$ and $c(\VE,P)=\exp\exp(-C\,\VE^{-1}\log\VE^{-1})$.
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.
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 $C_{k,\ell}^*+C\eps^{-2}$.