One example of a function that is Henstock-Kurzweil integrable but not Lebesgue integrable is $f(x) = \frac{1}{x} \cos\left(\frac{1}{x^2}\right)$ on $[0, 1]$. However, $f$ only fails to be locally-integrable around $0$. For any $\epsilon > 0$, the Lebesgue integral over $[\epsilon, 1]$ exists.
What can we say about the set of such points in general? In particular, we can define $$ S(f) = \{ x \in [0, 1] : \text{exists } \epsilon > 0 \text{ such that } f \text{ is HK-integrable on } [x - \epsilon, x + \epsilon] \text{ but } |f| \text{ is not HK-integrable on } [x - \epsilon, x + \epsilon]\}. $$ I want to know how big $S(f)$ can be, i.e., could it have an accumulation point, could it be co-meager, positive measure, etc?
Clearly, we can get any finite set by adding translates of the previous example. It's natural to try something like $\sum_n 2^{-n} f(x - x_n)$ for a dense set $\{x_n\}$, but it's difficult to show the absolute value fails to be integrable somewhere.