Cilleruelo 2011 concentration points two three dimensional modular

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corollary_1: A bound for points of xy = lambda mod p in a square box of side M valid for every M up to p, uniform in the shifts, which improves the bound of Chan and Shparlinski.

corollary_2: The product set of three intervals in the nonzero residues modulo a large prime, each of length less than p^{1/8}, is nearly as large as the product of their lengths.

corollary_3: Bounds the number of points of the exponential curve y = a g^x mod p in a square box of side M below the order of g, uniformly in the shifts; the count is at most M^{1/2+o(1)} when M is at most p^{1/3}.

corollary_4: Bounds the number of points of the exponential curve y = a g^x mod p in a square box of side M below the order of g; the count is at most M^{1/3+o(1)} when M is at most a constant times p^{1/8}.

theorem_1: Cilleruelo and Garaev's bound, uniform in the shifts, on the number of points of the modular hyperbola xy = lambda mod p in a square box of side M; it is M^{o(1)} once M < p^{1/4}.

theorem_2: Cilleruelo and Garaev's bound on the number of points of the three-dimensional modular hyperbola xyz = lambda mod p in a cube of side M: it is M^{o(1)} for M at most a constant times p^{1/8}, uniformly in the shift.

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Javier Cilleruelo, Moubariz Z. Garaev, Concentration of points on two and three dimensional modular hyperbolas and applications. Geometric and Functional Analysis 21 (2011), 892–904. arXiv:1007.1526, doi:10.1007/s00039-011-0127-6.

For a large prime p the authors bound I_2(M;K,L), the number of solutions of xy = lambda mod p in a box of side M, and I_3(M;L), the number of solutions of xyz = lambda mod p in a cube of side M. Theorem 1 proves I_2(M;K,L) < M^{4/3+o(1)}/p^{1/3} + M^{o(1)}, and M^{3/2+o(1)}/p^{1/2} + M^{o(1)} when K = L; in particular I_2 is M^{o(1)} once M < p^{1/4}, improving bounds of Chan and Shparlinski that relied on Bourgain's sum-product estimate, with Corollary 1 giving I_2 << M^2/p + M^{4/5+o(1)}. The proof of Theorem 1 follows an idea of Heath-Brown and rests on Lemma 1, that for m >= sqrt n the interval [m, m + n^{1/6}] contains at most two divisors of n. Theorem 2 handles the harder three-variable count by connecting it to the Pell equation, giving I_3(M;L) << M^{o(1)} for M << p^{1/8}, from which Corollary 2 yields |I_1 I_2 I_3| = (|I_1||I_2||I_3|)^{1-o(1)} for intervals shorter than p^{1/8}, and Corollaries 3 and 4 improve concentration bounds on exponential curves. Section 6 (pp. 11--12) poses Conjectures 1--4 and Problems 1--3, asking in particular for larger exponents than 1/4, 1/3 and 1/8 in the ranges of Theorems 1 and 2.

Source: https://arxiv.org/abs/1007.1526. The copy read for this card is arXiv:1007.1526v2, dated 12 Oct 2010 and titled "Concentration points on two and three dimensional modular hyperbolas and applications", not the journal article; the labels below are that preprint's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1007.1526), every other right reserved.

Bears on. #158: no result of the paper bears on the problem's question. Theorem 1 (p. 2) and Theorem 2 (p. 3) count points of xy = lambda and xyz = lambda modulo a prime with the variables in intervals of one length M.

Results. Labels and pages are those of arXiv:1007.1526v2 (pp. 1--12). Read status: claims checked for each page below; the proofs were read but not checked step by step.

• Theorem 1 (p. 2; proof Section 2, pp. 3--5): uniformly in K and L, I_2(M;K,L) < M^{4/3+o(1)}/p^{1/3} + M^{o(1)}, and I_2(M;L,L) < M^{3/2+o(1)}/p^{1/2} + M^{o(1)}; in particular I_2(M;K,L) < M^{o(1)} when M < p^{1/4}. The proof uses Lemma 1 (p. 3): for every positive integer n and m >= sqrt n, the interval [m, m + n^{1/6}] contains at most two divisors of n.
• Theorem 2 (p. 3; proof Sections 3--4, pp. 5--10): if M << p^{1/8}, then uniformly in L, I_3(M;L) << M^{o(1)}, through a connection with the Pell equation (Proposition 1, p. 5).
• Corollary 1 (p. 2; proof p. 10): uniformly in K and L, I_2(M;K,L) << M^2/p + M^{4/5+o(1)}, and I_2(M;L,L) << M^2/p + M^{3/4+o(1)}, improving Chan and Shparlinski.
• Corollary 2 (p. 3; proof p. 11): for intervals I_1, I_2, I_3 in F_p^* of length less than p^{1/8}, |I_1 I_2 I_3| = (|I_1||I_2||I_3|)^{1-o(1)}.
• Corollary 3 (p. 3; proof p. 10): for g >= 2 of multiplicative order t and M < t, uniformly in K and L, J_a(M;K,L) < (1 + M^{3/4} p^{-1/4}) M^{1/2+o(1)}.
• Corollary 4 (p. 3; proof pp. 10--11): in the same setting, J_a(M;K,L) < (1 + M p^{-1/8}) M^{1/3+o(1)}.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.