Higher BMI (Body Mass Index) is correlated with shorter sleep time. It is also correlated with later sleep than normal (going to bed late and getting up late.) Not enough sleep results in a tendency to gain weight. This isn’t true for everybody (some people actually lose weight when they don’t sleep enough), but over the population as a whole it has repeatedly been shown that less sleep results in added fat.
Sleep debt increases the appetite, and particularly increases the desire for high-calorie, carbohydrate-rich foods. Functional MRI scans show higher activation in an area of the brain involved with appetite. This is one obvious reason for weight gain, although the deeper picture is more complex. In some situations, researchers have found sleep restricted subjects did not have an increased appetite, but they still gained weight.
Although even one night of short sleep can result in a desire to eat more, the second night of a very short sleep left participants in a controlled study reaching for cookies and cake and eschewing fruit and vegetables.
Why the connection between sleep duration and excessive weight? When you are tired from insufficient sleep, you might fidget less and burn fewer calories, but a more likely explanation is the effect of sleep deprivation on hormones.