You can cover the area of sprouts with a heavy gauge woven geotextile, and then either mulch or spread large gravel or decomposed granite over the top of the geotextile. Or if the grass has thinned too much, you might plant Asiatic Jasmine groundcover, and use hedge trimmers to trim the jasmine and oak sprouts to a uniform height. You may choose to mow them along with the grass, if grass still exists.

When choosing an oak in a garden center, if there are sprouts coming up at the inside edges of the container, I would avoid that tree. Also, when roots are disturbed and damaged by rototilling, they are more likely to sprout suckers. Oaks having a slight tendency to sprout suckers will often do so when roots hit a barrier, such as trees confined to a parking lot planter, or between a sidewalk and driveway. But like freckles and sunshine, some trees have the ability to sucker, but do not unless stimulated to do so. It is a genetic trait, like freckles, except I like freckles. Only a small percentage of oaks send up suckers from the roots. What you see are sprouts from the roots of the existing tree therefore, you do not want to spray a herbicide on them in an attempt to eliminate the sprouts for you will hurt the “mother” tree along with the sprouts. If there were, you would be able to pull them up easily. There are almost never seedlings growing from acorns.

Comin out strong vine