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Butter and line a 23cm loose-bottomed tin with baking parchment. Put the digestive biscuits in a plastic food bag and crush to crumbs using a rolling pin, or blitz in a food processor. Transfer to a bowl, then pour over the melted butter. Mix thoroughly until the crumbs are completely coated. Tip them into the prepared tin and press firmly down into the base to create an even layer. Chill in the fridge for 1 hr to set firmly.
To make the filling, place the cream cheese, icing sugar and lime zest in a bowl, then beat with an electric whisk until smooth. Tip in the gin and double cream, and continue beating until the mixture is thick and completely combined. Now spoon it onto the biscuit base, and spread to the edges. Smooth the top of the cheesecake down with the back of a dessert spoon or spatula. Leave to set in the fridge overnight.
Pour the can of gin and tonic into a pan, add the caster sugar and lemon juice and boil hard. Once it has reduced to a syrup (about 3-4 minutes over high heat), cool a little, then add the lime juice and the slices of lemon and toss in the syrup. Allow to cool completely.
To serve, place the base of the cheesecake on top of a can, then gradually pull the sides of the tin down. Slip the cake onto a serving plate, removing the lining paper and base. Top with a handful of the crushed meringues, then drizzle with the syrup, scattering the lemon slices on top as you go. Add a final dusting of lime zest, slice and serve.