Proof of the FiniteField SumProduct Theorem

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Proof of the Finite-Field Sum-Product Theorem

• Boaz BarakAdditive Combinatorics Mini-Course

Princeton University
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The Sum Product Theorem
Overview of Proof
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Overview of Proof
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Plan

• State sum-product thm
• Overview of proof
• Lemma 1 (AA,A A small ? r(A) small)
• Lemma 2 (9 r, r(A) large)
• Two useful tools
• PR Lemma (Iterated Sums Lemma)
• BSG Lemma (Many representations Lemma)
• Proof of Lemma 1
• Proof of Lemma 2

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Two Useful Tools
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Two Useful Tools
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Proof of PR Lemma
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Proof of BSG Lemma
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Recap Two Tools
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Plan

• State sum-product thm
• Overview of proof
• Lemma 1 (AA,A A small ? r(A) small)
• Lemma 2 (9 r, r(A) large)
• Two useful tools
• PR Lemma (Iterated Sums Lemma)
• BSG Lemma (Many representations Lemma)
• Proof of Lemma 1
• Proof of Lemma 2

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Overview of Proof
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Overview of Proof
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Overview of Proof