Learn German: Es gibt viele Teilnehmer ohne Smartphone. - There are many participants without a smartphone.

The phrase es gibt (there is / there are) is a fixed expression. The grammatical subject is the singular pronoun es, which is why the verb is conjugated as gibt. The phrase viele Teilnehmer is actually the direct object.

Since viele Teilnehmer is the direct object, what case is it in?

It is in the accusative case! The expression es gibt always requires the accusative. Because Teilnehmer is plural, the accusative looks exactly like the nominative, but remembering this rule is crucial when dealing with masculine singular nouns.

How do we know that Teilnehmer is plural here?

Most masculine and neuter nouns that end in -er do not change their spelling in the plural. You have to look at the surrounding words to tell the difference. Here, the plural word viele (many) makes it clear that we are talking about multiple participants.

Why is there no article before Smartphone? Shouldn't it be ohne ein Smartphone?

While ohne ein Smartphone is grammatically correct, German often drops the indefinite article (ein / eine) after the preposition ohne. This is very common when talking about things in a general sense or as a category, rather than a specific object.

We're learning about nominalized infinitives (verbs turned into nouns). Is an example of this?