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This is the Past Exam Paper of Math Tripos which includes Restriction and Kakeya Phenomena, Minkowski Dimensions, Real Numbers, Besicovitch Set, Besicovitch Subset, Absolute Constant, Measurable Function, Discrete Fourier Analysis etc. Key important points are: Restriction and Kakeya Phenomena, Minkowski Dimensions, Real Numbers, Besicovitch Set, Besicovitch Subset, Absolute Constant, Measurable Function, Discrete Fourier Analysis, Fair Boolean Function
Typology: Exams
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Tuesday 3 June 2003 1.30 to 4.
Attempt ONE question from each of sections A, B and C
There are six questions in total. The questions carry equal weight.
The notation o(1) refers to a quantity which tends to zero as some other quantity (which will always be obvious from the context) tends to infinity.
Section A
1 Let E ⊆ Rn. Define the upper and lower Minkowski dimensions of E. Let Q be the set of all real numbers between 0 and 1 whose base 9 expansions contain only the digits 2 , 5 and 7. Show that the upper and lower Minkowski dimensions of Q are both 1/2. What is meant by the term Besicovitch set? Show that any Besicovitch set in R^2 has upper and lower Minkowski dimension 2.
2 Let p be a prime. What is meant by a Besicovitch subset of Fnp? Show that there is
a Besicovitch subset of F^2 p with cardinality 12 p^2 (1 + o(1)). Show that there is an absolute constant c > 0 so that every Besicovitch subset of F^15 p has cardinality at least cp^9.
Section B
3 Show that there is an absolute constant C with the following property. If f : S^1 → C is any measurable function with ‖f ‖∞ = 1, and if R ≥ 2, then
‖ f dσ̂ ‖L (^4) (B(0,R)) ≤ C(log R)^1 /^4.
(Here σ refers to the usual measure on the circle S^1 , and B(0, R) is the ball of radius R about the origin in R^2 ). Sketch how such a result can be used to prove that any Besicovitch set in R^2 has upper and lower Minkowski dimension 2 (details are not required).
4 Let p be a prime of the form 4k + 3, so that −1 is not a square in Fp. What
is meant by the discrete paraboloid P ⊆ F^3 p? Determine dσ̂ (x 1 , x 2 , x 3 ), where σ is the normalised counting measure on P. Define R∗(2 → 4), and show that it is at most 10 (you may state and use, without proof, any facts concerning discrete functional analysis and discrete Fourier analysis that you may require). Now suppose that p is of the form 4 k + 1, so that −1 is a square in Fp. Show that P contains a line. Hence, or otherwise, show that R∗(2 → q) is not bounded as p → ∞ for any q < 4.
Section C
5 (i) Prove that there is a quite fair boolean function f : Fn 2 → { 0 , 1 }, all of whose influences are at most 10 log n/n. You should define all of the terms used in this part of the question.
(ii) State Beckner’s inequality. Let A ⊆ Fn 2 have cardinality bN/ 2003 c, where N = 2n. Show that if N > N 0 is sufficiently large then
∫
ξ:|ξ|≤ 2
A(ξ)^2 dξ <
ξ:|ξ|> 2
A(ξ)^2 dξ.
Paper 7
