Exponents in the local properties problem for difference sets have a gap at 2

TL;DR

The paper resolves a gap phenomenon at the quadratic threshold for the local properties problem in difference sets: for even k, at the threshold value ell = k^2/4, the minimum number of differences in an n-point set with the local property remains subquadratic, specifically g(n, k, k^2/4) = O(n^{2 - 2^{-29}}). It introduces a framework of k-configurations to encode equal-difference relations, classifies them as c-good or c-bad, and uses a Behrend-type random construction to build large sets with controlled difference counts while avoiding c-bad configurations. A core backbone argument—comprising a baseline bound, a stability step, and a huge-star case—shows that any c-good k-configuration certifies at most a fixed (quadratic-like) number of pairs, yielding the subquadratic upper bound with a constant gap to 2. The method extends to odd k with an analogous, though slightly more delicate, bound. Overall, the work demonstrates a constant-sized jump in the exponent at the quadratic threshold and highlights distinct behavior from the graph-case thresholds, with potential implications for related extremal arithmetic problems.

Abstract

We study the local properties problem for difference sets: If we have a large set of real numbers and know that every small subset has many distinct differences, to what extent must the entire set have many distinct differences? More precisely, we define $g(n, k, \ell)$ to be the minimum number of differences in an $n$-element set with the `local property' that every $k$-element subset has at least $\ell$ differences; we study the asymptotic behavior of $g(n, k, \ell)$ as $k$ and $\ell$ are fixed and $n \to \infty$. The quadratic threshold is the smallest $\ell$ (as a function of $k$) for which $g(n, k, \ell) = Ω(n^2)$; its value is known when $k$ is even. In this paper, we show that for $k$ even, when $\ell$ is one below the quadratic threshold, we have $g(n, k, \ell) = O(n^c)$ for an absolute constant $c < 2$ -- i.e., at the quadratic threshold, the `exponent of $n$ in $g(n, k, \ell)$' jumps by a constant independent of $k$.

Figures (20)

• Figure 1: A plot of the bounds we know on $g(n, k, \ell)$ in the regime where $\ell$ is quadratic in $k$, where the $x$-axis depicts the coefficient of $k^2$ in $\ell$ and the $y$-axis depicts the exponent of $n$ in the bound --- a point $(a, c)$ means that for $\ell \approx ak^2$ we have a bound of roughly $n^c$. The purple line represents lower bounds --- for $2/9 < a < 1/4$ the best lower bound comes from \ref{['eqn:fps-lower']} with $r = 3$, and for other values of $a$ the best lower bound comes from \ref{['eqn:das-interm-lower']}. The blue line represents upper bounds, which come from \ref{['eqn:das-interm-upper']}.
• Figure 2: A version of Figure \ref{['fig:previous-bounds']} incorporating Theorem \ref{['thm:main']} (the value of $c$ is not to scale). There is a 'gap' on the $y$-axis between $c$ and $2$ --- the exponent of $n$ in $g(n, k, \ell)$ can never lie in this range when $k$ is even.
• Figure 3: A star of size $8$, where we depict a difference equality $x_{i_1} - x_{i_2} = x_{i_3} - x_{i_4}$ by placing $x_{i_1}$, $x_{i_2}$, $x_{i_4}$, $x_{i_3}$ in a parallelogram. A star of size $2p$ then consists of $p$ pairs with the same midpoint, which is the reason for the name. The shaded parallelogram shows that this star certifies $(8, 3)$ and $(8, 4)$.
• Figure 6: An example of a minimal implication (we have ${*} = -{*_1} - {*_2} + {*_3} + {*_4}$) and the corresponding graph used to prove \ref{['item:signs-pm1']}, where every edge is labelled with the ordinary variable it corresponds to.
• Figure 7: Two overlapping $2$-full sets of difference equalities as in Lemma \ref{['lem:2-full-intersection']}, with $\mathcal{T}_1$ in purple and $\mathcal{T}_2$ in blue. Here $\mathcal{T}_1$ has $4$ equations on $9$ variables and $\mathcal{T}_2$ has $5$ equations on $11$ variables; their intersection has $3$ equations on $7$ variables, and their union has $6$ equations on $13$ variables.

Theorems & Definitions (60)

• Theorem 1.1
• Proposition 1.2
• Example 2.1
• Example 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:random-constr']}
• Lemma 4.1
• Lemma 4.2