The phrase "a sort of attention" is correct and usable in written English.
It can be used to describe a type or manner of attention that is not specific or definitive.
Example: "She gave him a sort of attention that suggested she was interested, but not fully committed."
Alternatives: "a kind of attention" or "a type of attention".
"Reddit is a sort of attention aggregator," he said.
"My research suggests that they really attend to their target, so we could be seeing a sort of attention blindness," said Dr Wisniewska, explaining that porpoises might be so intently focused on the one fish they are hunting that they ignore their surroundings while they pursue it into a net.
There is above all a sort of attention-deficit quality to Vice's output, shifting abruptly between tones and sensibilities.
Whatever promise this plot may have held is undermined by a sort of literary attention deficit disorder.
Like Nick Carraway, Scalia "wanted the world to be in uniform and at a sort of moral attention forever".
There is a destabilizing, unrelenting directness in Myles's writing, and "Afterglow" is like the "Just Kids" of dog books: a punk devotional, shot through with a sort of divine attention to material reality and a poet's associative leaps.
It's sort of a chicken-and-egg thing -- is it possible to re-cultivate a sort of longer attention spans and curiosity, specifically by making films more available...I feel like a city without a revival house isn't a city.
I wanted to write about LA the way Balzac did about Paris or Philip Roth did about New Jersey or Michael Ondaatje did about Sri Lanka as a place that was worthy of a sort of ordinary attention, an arena for normal life instead of some soul-sucking vacuum.
Cracking them open on an iPad, for example, encourages a sort of half-attention as dozens of other news sources vie for attention through notifications or just through a nagging sensation of FOMO.
A sort of Short- Attention-Span-Theater for those of us whose lips move while we read the main Mudd blog, Reel Mudd (hey, I didn't name it; these people are librarians – give 'em a break, already) throws a delightfully random spotlight on past happenings around the campus and gives us a three-dimensional introduction to many traditions of the past and their exponents.
This entire book is illuminated by an unillusioned intelligence and a steady sort of attention to deeper, longer-range trends rather than what might be called headline-news ephemera.
