Hexagon

math.CO — Combinatorics

Square-difference-free sets of exponent 0.7580758

Contributed by Eric Naslund

Let $D(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no two elements whose difference is a nonzero perfect square. We give a computer-assisted construction proving $D(N)\ge N^{0.7580758318008816-o(1)}$ , improving the exponent $0.752796455874514\ldots$ of Krachun's ranked-block construction. The residues we use carry subintervals of $[0,1]$ that are ordered along every modular square difference. A finite stopping-word argument shows that if finitely many such alphabets, in pairwise coprime perfect-square moduli, have interval moments whose contributions sum to more than $\alpha$ , then $\alpha$ is an attainable exponent. Krachun's prime chains reproduce his exponent in this framework. The gain comes from an alphabet modulo a power of $4$ , built by a $25$ -state recursion, and from two composite alphabets, at the primes $5,43$ and at $19,23$ , in which the first differing digit is read separately at each prime. The numerical conclusion is computer-assisted. The theorem, including every finite certificate, is formalized in lean, PALOMAR-2026-09-19-000006 v2.

Primarily AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00135v1

math.NT — Number Theory

Improved sum-difference inequalities in abelian groups

Contributed by Logan Kleinwaks

For all finite subsets $X,Y$ of an abelian group, we prove \[ |X-Y|\le |X+Y|^{\lamI},\qquad \lamI=\frac{9451\e-3286}{5378\e+787}=1.454277448906\ldots, \] improving the classical exponent $3/2$ . We also prove that a universal two-set exponent $\lambda$ over the integers implies an upper bound $2-1/\lambda$ for the Gyarmati--Hennecart--Ruzsa constant, the supremum $\theta^*$ of $t$ such that $|A-B|\gg|A+B|^t$ and $|A+B|\ll|A|$ for arbitrarily large $A,B\subset\Z$ . Consequently, \[ \theta^*\le\frac{13524\e-7359}{9451\e-3286}=1.312373302115\ldots, \] improving their bound $4/3$ . The first argument combines a coupling with distinct differences, entropy inequalities for independent sums, a finite certificate using non-Shannon inequalities, and an explicit limiting certificate with weights $\int_0^1t(1-t)^k\e^t\,dt$. The transfer uses localisation and the Pl\"unnecke--Ruzsa inequality over a sequence of scales. The proofs are formalised in Lean~4.

Primarily AI-generated textHuman understanding: some partsShannon entropydifference setsformal verificationnon-Shannon inequalitiessumsets

Oct 07, 2026PDF ↗View work →

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Square-difference-free sets of exponent 0.7580758

Improved sum-difference inequalities in abelian groups

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