The uncertainty principle in finite abelian groups reveals a fundamental trade-off between a set's size and its Fourier transform. This powerful tool helps us understand the structure of sets with special additive properties, like small doubling.
Applications of the uncertainty principle are far-reaching in additive combinatorics. It's used to derive bounds on sum and difference sets, prove the existence of large Fourier coefficients, and plays a key role in important results like the Bogolyubov-Ruzsa lemma.
Uncertainty Principle for Finite Abelian Groups
Statement and Proof
- State the uncertainty principle for finite abelian groups
- For any non-empty subset A of a finite abelian group G, the product of the size of A and the size of its Fourier transform is at least the size of G
- Prove the uncertainty principle using the following tools:
- Orthogonality relations for characters of finite abelian groups
- Cauchy-Schwarz inequality
- Interpret the uncertainty principle as a quantitative version of the fact that a function and its Fourier transform cannot both be simultaneously localized
Fundamental Result and Applications
- Recognize the uncertainty principle as a fundamental result in additive combinatorics
- Apply the uncertainty principle to numerous problems in additive combinatorics, such as:
- Studying sum sets, difference sets, and related problems
- Deriving bounds on the size of sum sets and difference sets (see next section for more details)
- Establishing the existence of large Fourier coefficients for sets with small doubling (see later section for more details)
Uncertainty Principle Applications
Bounds on Sum Sets and Difference Sets
- Use the uncertainty principle to derive lower bounds on the size of:
- Sum sets A+B (where A+B = {a+b : a ∈ A, b ∈ B})
- Difference sets A-B (where A-B = {a-b : a ∈ A, b ∈ B}) for subsets A and B of a finite abelian group G
- Show that if A is a subset of G with small doubling (i.e., |A+A| ≤ K|A| for some constant K), then the size of the difference set A-A must be large
- More precisely, if |A+A| ≤ K|A|, then the uncertainty principle implies that |A-A| ≥ |G|/K
- Obtain similar bounds for the size of the sum set A+B in terms of the sizes of A, B, and their Fourier transforms
- Combine these bounds with other tools, such as the Plünnecke-Ruzsa inequalities, to study the structure of sets with small doubling or related additive properties
Applications in Additive Combinatorics Proofs
- Use the bounds derived from the uncertainty principle in conjunction with other tools to prove important results in additive combinatorics, such as:
- Freiman's theorem on sets with small doubling
- Bogolyubov-Ruzsa lemma (see later section for more details)
- Results on arithmetic progressions in sumsets
Large Fourier Coefficients
Existence of Large Fourier Coefficients
- Use the uncertainty principle to show that sets with small doubling must have large Fourier coefficients
- Specifically, if A is a subset of a finite abelian group G with |A+A| ≤ K|A| for some constant K, then there exists a non-trivial character χ of G such that the Fourier coefficient |Â(χ)| is at least |A|/√(K|G|)
- Prove this result using:
- The uncertainty principle
- The fact that the Fourier transform of the convolution 1A * 1A is equal to |Â|^2 (where 1A is the indicator function of A)
Applications in Additive Combinatorics Proofs
- Use the existence of large Fourier coefficients for sets with small doubling as a key ingredient in several proofs in additive combinatorics, such as:
- Freiman's theorem on sets with small doubling
- Bogolyubov-Ruzsa lemma (see next section for more details)
- Apply this result to study the structure of sets with small doubling and related additive properties
Uncertainty Principle vs Bogolyubov-Ruzsa Lemma
Statement of the Bogolyubov-Ruzsa Lemma
- State the Bogolyubov-Ruzsa lemma
- If A is a subset of a finite abelian group G with |A+A| ≤ K|A| for some constant K, then the 2k-fold sumset 2kA (where 2kA = A+A+...+A, k times) contains a non-trivial subgroup of G of size at least |G|/K^(O(k))
Proof of the Bogolyubov-Ruzsa Lemma
- Understand the role of the uncertainty principle in the proof of the Bogolyubov-Ruzsa lemma
- Use the uncertainty principle to find a non-trivial character χ with large Fourier coefficient |Â(χ)|
- Use this character to construct a dense subset of 2kA that is highly structured (i.e., a coset of a subgroup)
- Recognize the importance of the existence of large Fourier coefficients for sets with small doubling in the proof of the Bogolyubov-Ruzsa lemma
Applications of the Bogolyubov-Ruzsa Lemma
- Use the Bogolyubov-Ruzsa lemma as a powerful tool for studying the structure of sets with small doubling
- Apply the Bogolyubov-Ruzsa lemma to numerous problems in additive combinatorics, such as:
- Proving Freiman's theorem on sets with small doubling
- Studying arithmetic progressions in sumsets
- Investigating the structure of sets with related additive properties