[Paper Review] Dense sets of integers with prescribed representation functions

Javier Cilleruelo, Melvyn B. Nathanson|arXiv (Cornell University)|Aug 21, 2007

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper establishes that any integer-valued function $ f: \mathbb{Z} \to \mathbf{N} $ with $ \liminf_{|n|\to\infty} f(n) \geq g $ can be realized as the $ h $-fold representation function $ r_{\mathcal{A},h}(n) $ of a set $ \mathcal{A} \subset \mathbb{Z} $, constructed by enriching a $ B_h[g] $ sequence with a sparse auxiliary sequence. The key result is that such $ \mathcal{A} $ can be made as dense as $ \mathcal{A}(x) \gg \mathcal{B}(x\epsilon(x)) $, where $ \mathcal{B} $ is a $ B_h[g] $ sequence and $ \epsilon(x) \to 0 $, yielding $ \mathcal{A}(x) \gg x^{1/h - \varepsilon} $ for any $ \varepsilon > 0 $, matching the best known density for $ B_h[g] $ sequences.

ABSTRACT

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f (n)\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases "almost" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg x^{(1/h)-epsilon}.

Motivation & Objective

• To determine the maximal possible density of integer sets $ \mathcal{A} $ whose $ h $-fold representation function $ r_{\mathcal{A},h}(n) $ matches a prescribed function $ f(n) $ with $ \liminf_{|n|\to\infty} f(n) \geq g $.
• To establish a connection between the representation function problem and the construction of dense $ B_h[g] $ sequences, which are known to be difficult to construct with high density.
• To show that any such function $ f $ can be realized by a set $ \mathcal{A} $ whose counting function $ \mathcal{A}(x) $ grows as fast as $ \mathcal{B}(x\epsilon(x)) $, where $ \mathcal{B} $ is a $ B_h[g] $ sequence and $ \epsilon(x) \to 0 $.
• To improve upon Nathanson's earlier bound $ \mathcal{A}(x) \gg x^{1/(2h-1)} $ by achieving $ \mathcal{A}(x) \gg x^{1/h - \varepsilon} $ for any $ \varepsilon > 0 $, matching the best known lower bounds for $ B_h[g] $ sequences.

Proposed method

• The construction begins with a given $ B_h[g] $ sequence $ \mathcal{B} $, which ensures that initial representation counts are bounded by $ g $.
• A sparse auxiliary sequence $ \mathcal{U} = \{u_k\} $ is added incrementally to $ \mathcal{B} $, with elements chosen so that they increase the representation count at specific target values $ z_k $, where $ f(z_k) > r_{\mathcal{A}_{k-1},h}(z_k) $.
• The sequence $ \mathcal{U} $ is built using a Zeros Inserting Transformation $ T_\gamma^r $, which maps elements of $ \mathcal{B} $ into a new set with controlled density and sparsity.
• The representation function is maintained by ensuring that new representations involving $ u_{2k-1}, u_{2k} $ do not interfere with existing ones, using a recursive argument based on the number of summands $ h-s-t $.
• The density of the final set $ \mathcal{A} $ is bounded below by $ \mathcal{B}(x\epsilon(x)) $, where $ \epsilon(x) \to 0 $, by choosing the transformation parameter $ \gamma $ such that $ 2^{-2r\gamma^{-1}(\log_2 x)} \geq \epsilon(x) $.
• The proof proceeds by induction on the stage $ k $, showing that at each step $ r_{\mathcal{A}_k,h}(n) \leq f(n) $ for all $ n $, and that $ r_{\mathcal{A}_k,h}(z_k) $ increases to $ f(z_k) $ when needed.

Experimental results

Research questions

• RQ1 Can every function $ f: \mathbb{Z} \to \mathbf{N} $ with $ \liminf_{|n|\to\infty} f(n) \geq g $ be realized as the $ h $-fold representation function of some integer set $ \mathcal{A} $?
• RQ2 What is the maximal possible density of such a set $ \mathcal{A} $, in terms of the counting function $ \mathcal{A}(x) $, for a given $ f $?
• RQ3 How does the density of $ \mathcal{A} $ relate to the density of a $ B_h[g] $ sequence used as a base?
• RQ4 Can the construction achieve density close to the theoretical lower bound $ x^{1/h - \varepsilon} $, matching known bounds for $ B_h[g] $ sequences?

Key findings

• Any function $ f: \mathbb{Z} \to \mathbf{N} $ with $ \liminf_{|n|\to\infty} f(n) \geq g $ is realizable as the $ h $-fold representation function $ r_{\mathcal{A},h}(n) $ of some integer set $ \mathcal{A} $.
• The set $ \mathcal{A} $ can be constructed so that its counting function satisfies $ \mathcal{A}(x) \gg \mathcal{B}(x\epsilon(x)) $, where $ \mathcal{B} $ is a $ B_h[g] $ sequence and $ \epsilon(x) \to 0 $ as $ x \to \infty $.
• For any $ \varepsilon > 0 $, there exists $ g = g(h, \varepsilon) $ such that for any $ f $ with $ \liminf_{|n|\to\infty} f(n) \geq g $, the set $ \mathcal{A} $ satisfies $ \mathcal{A}(x) \gg x^{1/h - \varepsilon} $.
• The construction matches the best known lower bound for $ B_h[g] $ sequences, which is $ \mathcal{B}(x) \gg x^{1/h - \varepsilon} $, as shown by Vu (2000).
• For $ h = 2 $, the result recovers and improves upon Nathanson's bound, achieving $ \mathcal{A}(x) \gg x^{\sqrt{2}-1+o(1)} $, matching Ruzsa's construction for Sidon sets.
• The method provides a general framework to realize arbitrary representation functions with near-optimal density, resolving an open problem from [1] and [10].