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)|≪ Nexp(−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)|≪ Nexp(−c'(log N)5/7/(loglog N)2/7), improving the exponent to 5/7 (with an intermediate stage at 7/10). On the structured component E2small(N;J(N)), an algebraic and spectral container hierarchy combined with a generating-polynomial large sieve reaches the exponent 1−o(1): |E2small(N;J(N))|≪κNexp(−(1/40960+o(1))(log N)/((loglog N)(logloglog 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 2Θ(J4) to exp(𝒪(J2loglog 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 E2small. Second, to bypass unlocalized diffuse divisor sheets without subset enumeration, we construct a semi-composite congruence sieve s=pq≡−1pmod12ua (p∈(Z9/10,Z], q≤ Q rough). Conditioning on nbmod Msmall yields CRT mutual independence while decoupling the modulus 12ua≤ 12Z1/4 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−4u2v−w=n and difference sets D(Kc).
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BibTeX
@misc{dahan2026composite,
title = {{Composite witness sieves, Kneser--Plancherel containers, and the exponent $5/7$ for the Erdős--Straus conjecture}},
author = {Benjamin Dahan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08717},
eprint = {2611.08717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08717},
}
RIS
TY - UNPB
TI - Composite witness sieves, Kneser--Plancherel containers, and the exponent 5/7 for the Erdős--Straus conjecture
AU - Benjamin Dahan
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08717
UR - https://arxiv.org/abs/2611.08717
AB - 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)|≪ Nexp(−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)|≪ Nexp(−c'(log N)^5/7/(loglog N)^2/7), improving the exponent to 5/7 (with an intermediate stage at 7/10). On the structured component E_2^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^small(N;J(N))|≪_κNexp(−(1/40960+o(1))(log N)/((loglog N)(logloglog 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(𝒪(J^2loglog 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^small. Second, to bypass unlocalized diffuse divisor sheets without subset enumeration, we construct a semi-composite congruence sieve s=pq≡−1pmod12ua (p∈(Z^9/10,Z], q≤ Q rough). Conditioning on nbmod M_small yields CRT mutual independence while decoupling the modulus 12ua≤ 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^2v−w=n and difference sets D(K_c).
ER -