Popular Differences and the Croot--Lev Half-Threshold Problem

Abstract: Let $A$ be a finite non-empty subset of an abelian group $G$, and let $r_A(d)=|{(a,a&#39;)\in A<sup>2:a-a&#39;=d}|$. Croot and Lev asked whether the pointwise half-threshold condition $r_A(d)\ge |A|/2$ for every $d\in A-A$ forces $A-A$ to be either a subgroup or a union of three cosets. We resolve this open problem in its sharp general form by identifying the essential obstruction: the statement is false in arbitrary abelian groups, but becomes true after excluding non-zero two-torsion. More precisely, if $G$ is two-torsion-free and the half-threshold condition holds, then either $A-A$ is a finite subgroup of $G$, or there are a finite subgroup $H\le G$ and elements $x,g\in G$ such that [ A=(x+H)\cup(x+g+H). ] The two-torsion-free hypothesis is essential: for every $r\ge1$ we construct $A\subseteq\F_2<sup>{2r+1}$ with $A-A=\F_2<sup>{2r+1}\setminus{t}$ such that every non-zero represented difference has exactly $|A|/2$ representations, giving genuine counterexamples to the Croot--Lev conclusion. The proof of the positive result combines a Kneser quotient reduction with Lev's formulation of Kemperman's critical-pair theory.

• The paper proves that in 2-torsion-free abelian groups, if every difference has at least |A|/2 representations, then A−A is either a finite subgroup or three cosets arising from a two-coset structure of A.
• Its Kneser and Kemperman reduction converts the pointwise popularity condition into an exact critical configuration, showing that nonzero quotient differences occur uniformly |A|/2 times before eliminating larger aperiodic cases.
• It constructs characteristic-two counterexamples from quadratic-form graphs, where every represented nonzero difference has exactly |A|/2 representations, demonstrating that the torsion-free hypothesis is essential and leaving mixed-torsion classifications open.

The problem and the main result

Let $A$ be a finite non-empty subset of an abelian group $G$, with difference set $D = A - A$ and representation function $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$. Croot and Lev posed, in their open-problem collection on additive combinatorics [math/0611917], whether the pointwise half-threshold condition

$r_A(d) \ge |A|/2 \quad \text{for every } d \in D$

forces $D$ to be either a subgroup of $G$ or a union of three cosets of a subgroup. They had already observed that the strict version ($r_A(d) > |A|/2$ for all $d$) has a short proof: for any $x, y \in D$, the sets $G$0 and $G$1 both exceed half of $G$2, hence intersect, giving $G$3. The non-strict threshold is where the difficulty lies.

The paper resolves this problem in its sharp general form. The main theorem states that if $G$4 is 2-torsion-free (i.e., $G$5) and the half-threshold condition holds, then either $G$6 is a finite subgroup of $G$7, or there exist a finite subgroup $G$8 and elements $G$9 such that

$D = A - A$0

with the image of $D = A - A$1 in $D = A - A$2 having order different from $D = A - A$3, $D = A - A$4, and $D = A - A$5. In the second case,

$D = A - A$6

and a converse remark shows this alternative is attained exactly at the half threshold: differences in $D = A - A$7 have $D = A - A$8 representations while those in $D = A - A$9 have exactly $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$0. For finite groups, 2-torsion-free is equivalent to odd order, so the result directly answers the original question in odd-order groups. Crucially, the torsion hypothesis is not an artifact: the paper constructs genuine counterexamples in characteristic two, showing that the statement as posed by Croot and Lev is false in arbitrary abelian groups.

The Kneser reduction

The proof begins with an elementary counting consequence of the popularity hypothesis. Since $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$1 and $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$2, the condition yields $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$3, placing $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$4 in Kneser's critical range.

Writing $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$5 and $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$6 in $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$7, with $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$8 and $r_A(d) = |\{(a,a') \in A^2 : a - a' = d\}|$9, the reduction theorem establishes three facts, independent of parity:

• Exact critical size: $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$0 is aperiodic in $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$1 and $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$2. This follows by combining Kneser's lower bound $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$3 with the counting upper bound $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$4.
• Few holes: writing $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$5 for the number of holes of $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$6 inside the $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$7-cosets it meets, one obtains $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$8.
• Uniform representation at the threshold: if $r_A(d) \ge |A|/2 \quad \text{for every } d \in D$9, then $D$0 is even and $D$1 for every non-zero $D$2. The key mechanism is that the pointwise bound lifts through the fibres: each quotient representation accounts for at most $D$3 representations below, so $D$4, forcing $D$5; since the average over the $D$6 non-zero differences is exactly $D$7, equality must hold everywhere.

The consequence is that only two outcomes survive: $D$8 gives the subgroup case $D$9, and $G$0 gives the three-coset structure of $G$1. A further proposition upgrades the $G$2 case from a classification of $G$3 to a classification of $G$4: the fibre argument shows $G$5 and then $G$6, so $G$7 is exactly two full cosets. Thus any remaining obstruction must be a large aperiodic quotient whose non-zero differences are all represented exactly $G$8 times; ruling this out under the torsion hypothesis is the analytic core of the paper.

Ruling out uniform critical quotients

The elimination step uses Lev's formulation of Kemperman's critical-pair theory [math/0508179]. If $G$9 is a minimal counterexample with $r_A(d) > |A|/2$0 aperiodic, $r_A(d) > |A|/2$1, and $r_A(d) > |A|/2$2 for all non-zero $r_A(d) > |A|/2$3, then since $r_A(d) > |A|/2$4, Lev's reduction applies to the self-opposite pair $r_A(d) > |A|/2$5: there is a proper subgroup $r_A(d) > |A|/2$6 such that $r_A(d) > |A|/2$7 is elementary in $r_A(d) > |A|/2$8, where $r_A(d) > |A|/2$9.

Three cases are dispatched. If $d$0, minimality is contradicted by descending into $d$1. If some non-zero $d$2 has a unique quotient representation, a fibre argument forces the two fibres above its endpoints to be disjoint sets of size $d$3 covering $d$4; translating one against the other then shows each fibre is stabilized by its own difference set, hence is a coset of a subgroup $d$5, making $d$6 $d$7-periodic — contradicting aperiodicity unless $d$8 is trivial, which gives $d$9. Finally, Kemperman type IV would require a subgroup of order $x, y \in D$0 in the quotient, i.e., an element of order two — impossible when $x, y \in D$1 inherits 2-torsion-freeness from $x, y \in D$2 via the elementary lemma that quotients and subgroups of 2-torsion-free groups remain 2-torsion-free. For odd-order quotients the same contradiction arises because $x, y \in D$3 is even. This completes the classification.

Counterexamples in characteristic two

The positive theorem fails without the torsion hypothesis, and the failure is exhibited by quadratic forms over $x, y \in D$4. Let $x, y \in D$5, let $x, y \in D$6, set $x, y \in D$7, and take the graph

$x, y \in D$8

Then $x, y \in D$9 and $G$00. The verification rests on the derivative identity $G$01, where $G$02 is a non-zero balanced linear functional whenever $G$03; consequently every non-zero represented difference has exactly $G$04 representations, so the half-threshold condition holds. Yet $G$05 is odd and exceeds $G$06, so it can be neither a subgroup of the elementary 2-group nor a union of at most three cosets of a common subgroup. The smallest instance ($G$07) is the four-point set $G$08 in $G$09, whose difference set is the cube with one vertex removed. A product construction with an arbitrary finite abelian group $G$10 lifts these counterexamples to infinite families in groups of the form $G$11, since the representation function factors as $G$12.

These examples show that the dichotomy "subgroup or three cosets" is genuinely false as stated in the Croot–Lev problem, and that 2-torsion is the precise obstruction: the type IV case of Kemperman's classification, which is harmless in odd-order quotients, becomes realizable over characteristic two.

Limitations and open questions

The classification is complete for 2-torsion-free groups, but the paper leaves the mixed-torsion case open: it does not classify which configurations arise in groups having both odd-order components and non-trivial 2-torsion beyond the product counterexamples given. The counterexample family is tied specifically to non-degenerate quadratic forms in characteristic two; whether every counterexample to the Croot–Lev conclusion must arise from such a construction, or can be reduced to one via the Kneser quotient machinery, is not addressed. It also remains open whether analogous thresholds other than $G$13 admit comparable structural classifications, and whether the two-coset conclusion can be sharpened to control the order of $G$14 modulo $G$15 beyond excluding $G$16, $G$17, and $G$18.

Conclusion

This paper settles the Croot–Lev half-threshold problem with a sharp dichotomy: in 2-torsion-free abelian groups, universal half-popularity of differences forces $G$19 to be a subgroup or a three-coset set arising from a two-coset configuration of $G$20 itself, while explicit quadratic-form constructions show the statement fails in characteristic two for every dimension. Methodologically, the argument demonstrates how a pointwise representation hypothesis can be converted, via a Kneser stabilizer quotient, into a uniform critical-pair condition amenable to Kemperman-type structure theory — a reduction that may prove useful for related inverse problems concerning popular sums and differences.