Introduces δ(G,H) for finite abelian quotients, proves δ(G,H) ≥ 2|G/H| - m(G,H) sharp for cyclic cases, and conjectures δ=(2p-1)² for the (Z/p²Z)² case with lower bound 3p²-p-1.
A sum-product estimate in finite fields, and applications
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abstract
Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.
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math.NT 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Transversal Difference Numbers in Finite Abelian Quotients
Introduces δ(G,H) for finite abelian quotients, proves δ(G,H) ≥ 2|G/H| - m(G,H) sharp for cyclic cases, and conjectures δ=(2p-1)² for the (Z/p²Z)² case with lower bound 3p²-p-1.