The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number theorist, so please bear with me.

Let $c $ be a positive real number, let $\theta$ be an irrational real number, and let $N$ be a positive integer. For real $r$, define $d(r)$ to be the distance from $r$ to the integers (the norm symbol, $\| r\|$, is often used to denote this, causing confusion to non-number theorists like me).

Define subsets of the set of integers in the interval $(N/2, N/2 + \sqrt N)$, $$ T_{N,c, \theta} = \left\{k \left| N/2 \leq k \leq N/2 + \sqrt N, \&\ d\left({N\choose k}\theta\right) > c \right.\right\}. $$

My question: is it true that for every irrational $\theta$, there exists $c$ (depending on $\theta$), such that $$ \limsup_{N\to \infty} \frac{\left|T_{N,c,\theta}\right|}{\sqrt N} > 0? $$ That is, do there exist infinitely many $N$ such that for some $c, \eta> 0$, $|T_{N,c,\theta} |\geq \eta \sqrt N$ (both $\eta$ and $c$ are permitted to depend on $\theta$)?

Unfortunately, "almost all"-type results (e.g., the set of irrational $\theta $ for which this doesn't hold is of measure zero) are not useful for what I'm interested in.

Or is this currently impossible to decide?