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While this conjecture is false for even , Beck has proved that  always holds with probability . Recently, He, Ju\u0161kevi\u010dius, Narayanan, and Spiro conjectured that the Erd\u0151s' conjecture holds when  is odd. We disprove this conjecture by exhibiting vectors  for which  occurs with probability . On the other hand, a relaxed version of their conjecture holds: we show that we always have  with probability , for all . This shows that when  is odd, the minimum probability that  exhibits a double\u2010jump phase transition at , as we can also show that  occurs with probability at least  for some . Additionally, and using a different construction, we give a negative answer to a question of Beck and two other questions of He, Ju\u0161kevi\u010dius, Narayanan, and Spiro, concerning the optimal constructions minimising the probability that . We also make some progress on the higher dimensional versions of these\u00a0questions.<\/jats:p>","DOI":"10.1112\/jlms.70539","type":"journal-article","created":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T05:41:13Z","timestamp":1778823673000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Double\u2010jump phase transition for the reverse Littlewood\u2013Offord problem"],"prefix":"10.1112","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-7904-1256","authenticated-orcid":false,"given":"Lawrence","family":"Hollom","sequence":"first","affiliation":[{"name":"Institute of Mathematics \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne Lausanne Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5360-6917","authenticated-orcid":false,"given":"Julien","family":"Portier","sequence":"additional","affiliation":[{"name":"Institute of Mathematics \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne Lausanne Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7598-7792","authenticated-orcid":false,"given":"Victor","family":"Souza","sequence":"additional","affiliation":[{"name":"Department of Mathematics Cornell University Ithaca New York USA"},{"name":"Sidney\u2010Sussex College Cambridge UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,5,14]]},"reference":[{"key":"e_1_2_12_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(83)80002-X"},{"key":"e_1_2_12_3_1","doi-asserted-by":"publisher","DOI":"10.4169\/amer.math.monthly.120.04.362"},{"key":"e_1_2_12_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(81)90085-9"},{"key":"e_1_2_12_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2013.04.007"},{"key":"e_1_2_12_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85221-6_1"},{"key":"e_1_2_12_7_1","doi-asserted-by":"publisher","DOI":"10.55016\/ojs\/cdm.v6i1.62011"},{"key":"e_1_2_12_8_1","doi-asserted-by":"publisher","DOI":"10.1002\/047174882X"},{"key":"e_1_2_12_9_1","series-title":"Proc. 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