Moving cautiously, since the departure of his Native American support had deprived him of reliable reports on the American position, Burgoyne advanced to the south after crossing the Hudson. On September 18 the vanguard of his army had reached a position just north of Saratoga, about 4 miles (6.4 km) from the American defensive line, and skirmishes occurred between American scouting parties and the leading elements of his army.

Can we conclude that the battle of saratoga 1777 took place near the hudson river?
yes