In a medium saucepan over medium heat, combine the sugar and 1 cup of the water. Heat until the sugar has just dissolved. Remove the pan from heat and cool the mixture to room temperature.
In a pitcher, add the lemon juice and room temperature or chilled syrup. Stir in the remaining 3 1/2 cups of cold water and the heavy cream. Add the vodka, if desired. Serve chilled over ice garnish with lemon slices, if desired.
This may be prepared ahead of time and stored chilled in the refrigerator.