Americans dream of summering in France, but saxophonist Brad Wheeler, who moved from Chicago to Paris several years ago, is spending the last part of his summer vacation in Illinois, where his depth of purpose and committed urgency always find a sweet home. Wheeler plays music as if he’s been thinking about nothing else since the last time he held his sax. Even when he fills a solo with skittering trails of notes, they carry weight and resolve, instead of fluttering off like the contrails of his more facile contemporaries. But then the word “facile” rarely comes up when discussing Wheeler. Some saxists work hard to make it sound easy, but Wheeler, intently focused on probing a given composition, doesn’t worry about whether some of the seams show. His improvisations, especially on tenor, sail along on a tough, full tone: layered, versatile, and an appealing throwback to the tough-and-tender expressiveness of the saxists who passed through the Miles Davis bands (Coltrane, Shorter, and especially Hank Mobley). During his Paris years Wheeler has become increasingly involved with writing and arranging. What impact have those activities had on his already compact and admirably structured improvisations? The best chance to find out comes Saturday, when Wheeler fronts a formidable band featuring bassist Eric Hochberg and pianist Laurence Hobgood, both of whom treat composition as seriously as Wheeler himself. Friday Wheeler sits in with the Green Mill All-Stars, along with two notable ringers: veteran drummer Wilbur Campbell and pianist Willie Pickens (of Elvin Jones’s band). Friday, 9 PM, and Saturday, 8 PM, Green Mill, 4802 N. Broadway; 878-5552.