Special equipment: Cocktail shaker
For the rosemary simple syrup: Combine the sugar, 3/4 cup water and the rosemary sprigs in a small saucepan over medium heat. Bring to a simmer and cook, stirring and muddling the rosemary with a wooden spoon, until the sugar has completely dissolved, about 1 minute. Take off the heat and cool to room temperature. Strain into a small container. The syrup can be kept refrigerated in an airtight container up to 2 weeks.
For the rosemary gin fizz: Combine the gin, lemon juice, lime juice and 2 tablespoons simple syrup in a cocktail shaker filled with ice. Shake vigorously, then strain into a rocks glass filled with ice. Top off with the seltzer and garnish with the rosemary sprig.