An exponential bound for exponential diffsequences

Abstract: A theorem of van der Waerden states that for any positive integer $r$, if you partition $\mathbf{N}$ into $r$ disjoint subsets, then one of them will contain arbitrarily long arithmetic progressions. It is natural to ask what other arithmetic structures are preserved when partitioning $\mathbf{N}$ into a finite number of disjoint sets and to pose quantitative questions about these. We consider $D$-diffsequences, introduced by Landman and Robertson, which are increasing sequences in which the consecutive differences all lie in some given set $D$. Here, we consider the case where $D$ consists of all powers of $2$ and define $f(k)$ to be the smallest $n$ such that partitioning $\{1,2,\cdots,n\}$ into $2$ subsets guarantees the presence of a $D$-diffsequence of length $k$ contained entirely within one subset. We establish that $f(k)$ grows exponentially.

number theory

Audience: researchers in the discipline

POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

Participants are welcome to stay following the talks to have further discussions with the speakers or meet other audience members.

To join our mailing list for talk reminders and other updates, send an email to NDNTseminar+join[AT ]g-groups[DOT]wisc[DOT]edu. Note: You will receive an automatic response prompting you to either click on a button, or to reply to that automatic email. Please use the latter option.