[Paper Review] On Larcher's theorem concerning good lattice points and multiplicative subgroups modulo p

Nikolay Moshchevitin, Дмитрий Михайлович Ушанов|ArXiv.org|Jun 23, 2009

Finite Group Theory Research5 references3 citations

TL;DR

This paper establishes the existence of well-distributed lattice points in thick multiplicative subgroups modulo a prime $ p $, leveraging character sum estimates and continued fraction theory. It proves that for sufficiently large subgroups $ R = v \cdot U $, at least half of the elements $ a \in R $ have all continued fraction coefficients $ b_i(a) < 16\log p $, and some $ a \in R $ satisfy $ \sum b_i(a) \ll \log p \log\log p $, implying low discrepancy $ D_p(a) \ll \log p \log\log p $.

ABSTRACT

We prove the existence of two-dimensional good lattice points in thick multiplicative subgroups modulo $p$.

Motivation & Objective

• To generalize Larcher’s result on good lattice points by extending it to multiplicative subgroups modulo a prime $ p $.
• To establish conditions under which elements $ a \in R = v \cdot U $, where $ U $ is a multiplicative subgroup, yield well-distributed points $ \left( \frac{x}{p}, \left\{ \frac{ax}{p} \right\} \right) $ in $[0,1]^2$.
• To bound the partial quotients in the continued fraction expansion of $ a/p $, ensuring they are $ \ll \log p $, which controls discrepancy.
• To show that under mild size conditions on $ \#R $, there exists $ a \in R $ such that the discrepancy $ D_p(a) \ll \log p \log\log p $.

Proposed method

• Use of character sum estimates over dyadic rectangles $ \Pi^t $ to control exponential sums related to the discrepancy of lattice points.
• Application of Burgess’s character sum bound with $ r=2 $ to derive $ \left| \sum_{(x,u)\in\Pi^t} \chi(x)\overline{\chi(u)} \right| \ll p^{7/8} \log^2 p / \sqrt{c} $, where $ c $ controls the rectangle size.
• Employing continued fraction theory: Lemma A ensures that good rational approximations to $ a/p $ are convergents, and Lemma B bounds the approximation error via partial quotients.
• Defining a function $ f_a(x) $ that counts how well $ \| ax/p \| $ is approximated, linking it to partial quotients $ b_i(a) $, and using partial summation to estimate $ \sum b_i(a) $.
• Using orthogonality of characters to express the number of solutions to $ ax \equiv y \pmod{p} $ in terms of character sums, and bounding these sums via Lemma 1.
• Combining subgroup structure with character sum bounds to show that a positive proportion of $ a \in R $ have all $ b_i(a) < 16\log p $, and some have $ \sum b_i(a) \ll \log p \log\log p $.

Experimental results

Research questions

• RQ1 Under what conditions on a multiplicative subgroup $ U \subset \mathbb{Z}_p^* $ does there exist $ a \in v \cdot U $ such that the discrepancy $ D_p(a) $ is small?
• RQ2 Can the partial quotients in the continued fraction expansion of $ a/p $ be uniformly bounded for a positive proportion of $ a \in R = v \cdot U $, and if so, what is the optimal bound?
• RQ3 What size condition on $ \#R $ ensures the existence of $ a \in R $ with $ \sum b_i(a) \ll \log p \log\log p $, implying low discrepancy?
• RQ4 How do character sum estimates over structured rectangles $ \Pi^t $ contribute to bounding discrepancy in lattice point sets modulo $ p $?

Key findings

• For any multiplicative subgroup $ U \subset \mathbb{Z}_p^* $, if $ \#R \geq 10^5 p^{7/8} \log^{3/2} p $ with $ R = v \cdot U $, then at least half of the elements $ a \in R $ have all continued fraction coefficients $ b_i(a) < 16\log p $.
• If $ \#R \geq 10^8 p^{7/8} \log^{5/2} p $, then there exists $ a \in R $ such that $ \sum_{i=1}^{l(a)} b_i(a) \leq 500 \log p \log\log p $, which implies $ D_p(a) \ll \log p \log\log p $.
• The discrepancy bound $ D_p(a) \ll \log p \log\log p $ is achieved for some $ a \in R $, improving upon previous results and matching the best-known bounds for general good lattice points.
• The character sum estimate in Lemma 1 provides a crucial bound: $ \left| \sum_{(x,u)\in\Pi^c} \chi(x)\overline{\chi(u)} \right| \leq 10000 p^{7/8} \log^2 p / \sqrt{c} $, which enables the main results.
• The proof uses a function $ f_a(x) $ that tracks how well $ \| ax/p \| $ is approximated, linking it to partial quotients via continued fraction theory and partial summation.
• The result is robust: the constants $ 10^5 $ and $ 10^8 $ in the size conditions are not optimal and may be reduced, though the asymptotic dependence on $ p $ and $ \log p $ is tight.