Composite witness sieves, Kneser--Plancherel containers, and the exponent $5/7$ for the Erdős--Straus conjecture

Abstract

For the Erdős--Straus equation $4/n=1/x+1/y+1/z$, Vaughan's unconditional bound on the exceptional set $E(N)$ has stood at $|E(N)|\ll N\exp(-c(\log N)^{2/3})$ since 1970. We harness composite divisor witnesses at growing depth to cross this $2/3$ barrier unconditionally. On the full exceptional set $E(N)$, we prove the unconditional bound $|E(N)|\ll N\exp(-c'(\log N)^{5/7}/(\log\log N)^{2/7})$, improving the exponent to $5/7$ (with an intermediate stage at $7/10$). On the structured component $E_2^{\mathrm{small}}(N;J(N))$, an algebraic and spectral container hierarchy combined with a generating-polynomial large sieve reaches the exponent $1-o(1)$: $|E_2^{\mathrm{small}}(N;J(N))|\ll_κN\exp(-(\frac{1}{40960}+o(1))\frac{\log N}{(\log\log N)(\log\log\log N)^κ})$, matching the CRT capacity scale. These results resolve the obstructions to composite-witness sieving at growing depth. First, we replace subgroup generation by Kneser's stabilizer $K(M)$, which embeds into a $2$-primary character kernel, reducing container counts from $2^{Θ(J^{4})}$ to $\exp(\mathcal{O}(J^{2}\log\log J))$. Coupling this with a generating-polynomial large sieve, Markov $1/2$-popular symmetry containers, and Pontryagin--Plancherel character containers yields the exponent $1-o(1)$ on $E_2^{\mathrm{small}}$. Second, to bypass unlocalized diffuse divisor sheets without subset enumeration, we construct a semi-composite congruence sieve $s=pq\equiv-1\pmod{12ua}$ ($p\in(Z^{9/10},Z]$, $q\le Q$ rough). Conditioning on $n\bmod M_{\mathrm{small}}$ yields CRT mutual independence while decoupling the modulus $12ua\le 12Z^{1/4}$ from $q$; a Bonferroni majorant and the Bombieri--Vinogradov theorem then yield the unconditional $5/7$ exponent on all of $E(N)$. Finally, we study bilinear dispersion on $4uavw-4u^{2}v-w=n$ and difference sets $D(K_c)$.