arXiv subjects
Publications and source records attributed to Alex Burgin.
Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective property with power savings (b) characterize uniform distribution, (c) establish polynomial mean ergodic theorems and (d) study metric pair correlation of Cantor sets.
Let $\mathcal C= \{k_1 0. $$ This is an extension of the IP Ergodic Theorem of Furstenberg and Katznelson, and a partial extension of recent work of Kra and Shalom. In particular, this implies that for any subset of integers $A$ of positive upper Banach density, there is a set $B$ of integers $n$ of positive lower Banach density such that $A$ contains an $\ell+1$ term progression, with step size $k_n$, where $n\in B$. This is a complement to recent results of Kra and Shalom, for IP Sets of integers, and Burgin, concerning Sarkozy's Theorem for Primes with restricted digits.
We study recurrence along shifted primes with restricted digits. By constructing a local approximant to the associated exponential sums, we prove the van der Corput property for shifted primes with restricted digits. This in turn shows that if $A\subset \mathbb{N}$ has positive upper Banach density, then there exists some prime $p$ with restricted digits and two elements $a_1,a_2\in A$ such that $a_1+p-1=a_2$.
We give a new elementary proof of Landau's Prime Ideal Theorem. The proof is an extension of Richter's proof of the Prime Number Theorem. The main result contains other results related to the equidistribution of the prime ideal counting function.
We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen arXiv:2205.04786, we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou arXiv:2208.02637 for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.