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.
Sum-product estimates for rational functions
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abstract
We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.
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math.NT 1years
2026 1verdicts
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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.