DWDS has many examples of the use of both "zu Arbeiten" and "zum Arbeiten", and both always appear to translate to "to work". So why are there 2 different ways of saying the same thing?
Here are long lists from DWDS of, first, zu Arbeiten, capital A:
and, second, zum Arbeiten:
zu arbeiten, small "a" is not in question here.
That piece zu Arbeiten has Arbeiten as the dative plural of the noun die Arbeit as an argument to the preposition zu. That adverbial makes only sense in a very narrow range of contexts.
Sein Werk steht in Beziehung zu Arbeiten seines Vaters.
His oeuvre has relations to works of his father.
While the piece zum Arbeiten has Arbeiten as the dative singular of the noun das Arbeiten made from the verb arbeiten. The meaning is similar to that of the English gerund, or sometimes, the infinitive.
Zum Arbeiten geht er in die Werkstatt.
For working he goes to the workshop.
Sometimes zu Arbeiten and zum Arbeiten become almost synonymous.
Man zwingt ihn zum Arbeiten.
They force him to work.
Man zwingt ihn zur Arbeit.
They force him to work on something.
Man zwingt ihn zu Arbeiten.
They force him to work on multiple things.
Zu arbeiten (without capital A) means to work in the sense of, e.g.,
I like to work.
(Ich arbeite gerne. / Ich mag es, zu arbeiten.)
Zum Arbeiten means to (the) work in the sense of, e.g.,
I came here to work.
(Ich komme zum Arbeiten her.)
Here, Arbeiten is a noun.
In “zu Arbeiten”, “Arbeiten” is the plural of “die Arbeit” with an indefinite article. In “zum Arbeiten” or “zu dem Arbeiten” we have the nominalisation “das Arbeiten” of The verb “arbeiten”.
