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Advice to Research Students
- A problem is something that you are approaching from the wrong direction.
- The original papers on a problem may contain good ideas that have been forgotten.
- Progress often comes from applying an idea from somewhere else to your problem.
- When studying a paper, ask the following questions.
- What is this about?
- Why is it done this way?
- Can't you use instead...?
- Isn't this the same as...?
- Notions are more important than notations, but bad notation can hold you back, whilst good notation can suggest connections with other problems.
- Crazy ideas sometimes work.
Some useful ideas
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- Most inequalities rest on the fact that a modulus squared is positive.
- If you squeeze an inequality too hard, it can become an equation. An integer with |n| < 1 is zero. A set of integer points in n dimensions occupying a volume smaller than constant times n must all lie on a plane.
- Dirichlet's pigeon-hole principle (compactness). If R letters are delivered to R-1 people, then some lucky person must have two or more letters. If R points lie in an interval, some pair of points must be close together.
- Given a set of R numbers (integers, rational numbers, algebraic numbers), let p be a prime < R. One pair of numbers must be congruent modulo p, and so cannot be close together.
- Size ranges: if a factor or denominator n lies in a range 1 to N, then it often helps to divide the range into blocks between consecutive powers of two.
- Divide and conquer. Divide a long range into short ranges. Approximate within each short range (maybe squeezing an inequality into an equation). Combine the results from the short ranges, for example by proving that not all short ranges can reduce in the same way.
- The Dirichlet interchange. If d, a factor of n, is getting too big, then work with the other factor e in de = n.
- The Riesz interchange. If different cases have different numbers of solutions, ask what it takes for a case to have more than R different solutions, and get a bound for how many cases can have at least R solutions.
- Averaging. Can you average over one of the variables instead of working things out for individual values?
- When averaging a sum of error terms, it sometimes helps to put in extra positive terms so your average will extend over all integers in the range, not just over a subset of them.