Differences between 'Can I help you?' and 'May I help you?'

[Janet~]

Dear all,
1.Are there differences between "Can I help you?" and "May I help you?"?
2.These folks don't work for big companies or universities with well-heeled R&D labs, fancy equipment, and helpful assistants. often they are people like Frampton Ellis, who in 1981 earned his first patent for an athletic shoe designed to cut down on ankle injuries by mimicking the bare foot.
Why cut down on? What's the function of ON? Can we just say cut down? This question has been troubled me for a long time. And also "look up at sky".Why at?
thank you in advance.
Janet