In a pitcher, combine the simple syrup, apple juice, rum, and sparkling water. To serve, fill a martini shaker with ice and add about 1 cup of the rum mixture. Shake well and pour into martini glasses. Garnish with apple slices and serve.
In a small saucepan, combine the sugar, water, and ginger over medium heat. Bring to a boil, reduce the heat, and simmer for 5 minutes, stirring occasionally, until the sugar has dissolved. Remove the pan from the heat and allow the syrup to cool, about 20 minutes. Strain before using.
Recipe courtesy of Giada De Laurentiis