MATSALAR DA'IRA TA GAUSS TA SAUKA ƘASA DA 5/16

Ɗauki da’ira wadda cibiyarta take kan ɗigon raga na wata takarda mai murabba’ai marar iyaka, ka ƙirga mahaɗar layukan da suka faɗa a cikinta. Amsar tana kusa da filin da’irar — amma ba ta taɓa yin daidai da shi ba, domin gefen yana yanke ragar ta hanyar da ba ta da tsari. Matsalar da’ira ta Gauss tana tambayar har yaya wannan rashin daidaito zai iya girma.

A cikin alamomin takardar, faifan shi ne a² + b² ≤ X, inda X shi ne murabba’in radius, don haka filinsa πX ne. Rashin daidaiton shi ne

E(X) = #{(a, b) ∈ ℤ² : a² + b² ≤ X} − πX

— adadin ɗigon raga (a, b) da ke ciki, a cire filin. Duk masu nunawa (exponents) da ke ƙasa ƙarfi ne na X, kamar a cikin takardar.

Daga kewaye zuwa hasashe

Hujjar siffa ta Gauss tana ba da E(X) = O(X^{1/2}): kuskuren bai wuce girman radius ba, kusan girman iyakar da’irar. Wannan ita ce iyaka mai sauƙi.

A ɗaya ƙarshen, Hardy ya tabbatar cewa kuskuren yana ci gaba da jujjuyawa, a hanyoyi biyu, zuwa ƙimomi masu girman X^{1/4} — don haka babu iyaka mai mai nunawa ƙasa da 1/4 da za ta iya tabbata. Amsar da ake tsammani, wadda takardar ke kira hasashen Hardy (Hardy’s conjecture), ita ce 1/4 kusan daidai ne: |E(X)| ≲ X^{1/4+ε} ga kowane ε > 0, inda ε ke nufin asara wadda ta fi kowane ƙarfi tsayayye ƙanƙanta.

Duk abin da ya biyo baya yaƙi ne a kan mai nunawa tsakanin 1/2 da 1/4. Teburin tarihin takardar ya jera matakan: Sierpiński a 1906 da 1/3, van der Corput, Titchmarsh, Hua, Chen, Kolesnik, Iwaniec da Mozzochi da 7/22 a 1988, Huxley da 23/73 a 1993 da 131/416 (kusan 0.3149) a 2003, sannan Li da Yang, a cikin wani preprint na 2023, da 0.3144831759…

Ƙofar a 5/16

Mafi yawan wannan ci gaban ya fito ne daga wata dabara da Bombieri da Iwaniec suka gina, Iwaniec da Mozzochi suka daidaita ta ga da’ira. Tana mayar da matsalar zuwa ƙiyasi a kan jimlolin exponential (exponential sums), waɗanda ke raguwa zuwa matsaloli biyu na “tazara” (spacing): ƙirga sau nawa wasu ƙimomin kusanci na lambobin rabo (rational approximations) suke faɗuwa kusa da juna. Huxley ya tura wannan hanya mafi nisa, kuma a 1993 ya ba da shawarar 5/16 = 0.3125 a matsayin iyaka mai yiwuwa ta hanyar Bombieri–Iwaniec kamar yadda take a lokacin. Tarihin Li da Yang, wanda aka samu ta hanyar kawo wani sabon kayan aiki daga nazarin harmonic — ka’idar “small cap decoupling” ta Guth da Maldague — har yanzu yana sama da ita.

Xi Cen (Jami’ar Ma’adinai da Fasaha ta China-Beijing), Xitao Gao da Junyong Zhang (Cibiyar Fasaha ta Beijing) da Cheng Zhang (Cibiyar Kimiyyar Lissafi ta Yau, Jami’ar Tsinghua) yanzu sun tabbatar:

Ka’ida ta 1.1. Ga kowane ε > 0 da kowane lamba ta gaske X ≥ 4, |E(X)| ≲ X^{209/670+ε}.

Kuma sun bayyana daidai nawa wannan ya sauka ƙasa da layin:

209/670 = 5/16 − 3/5360

Wannan kusan 0.31194 ne — riba ta 0.0025 a kan Li da Yang. Ƙarami a girma, amma a ɗaya gefen ƙimar da ta yi aiki a matsayin alama ga wata hanya gaba ɗaya.

Gyare-gyare biyu, zanguna biyu

Hujjar ta raba sauran ma’aunai zuwa zanguna biyu, waɗanda suke haɗuwa daidai a wurin da duka biyun ke ba da 209/670.

A zango na tsakiya, marubutan sun faɗaɗa wani lokaci (phase) zuwa mataki na shida kuma sun yi amfani da ka’idar matsakaicin ƙima ta Vinogradov don rage komai zuwa ƙirgar nau’ikan ɗigo waɗanda suka yarda a cikin ma’auni na huɗu, na biyar da na shida na Taylor. Muhimmin mataki shi ne warware waɗannan sharuɗɗa a fili: kuskuren layi (linear) yana sokewa, yana barin kuskuren murabba’i (quadratic) ƙarami isa a sarrafa shi ta faɗaɗawar Fourier da ƙiyasin tazara na Robert da Sargos. Ƙirgar nau’ikan ta fito ƙarama kamar nau’ikan “diagonal” marasa muhimmanci, har zuwa asara ƙanana yadda aka so.

A zango mafi girma, sabon sinadarin shi ne ƙiyasin rarrabawa ga jimlolin exponential waɗanda mitocinsu (frequencies) suke a kan mazugi (cone), an samo shi daga ƙiyasin Guth da Maldague mai dogaro da girman motsi (amplitude). Yana samar da sharuɗɗa biyu, ɗaya da ƙarfi na huɗu ɗaya kuma da ƙarfi na shida. Riƙe su daban ta cikin double large sieve, maimakon haɗa su zuwa ƙiyasin moment ɗaya da farko, shi ne abin da ya biya. A cewar takardar, wannan yana aiki ba tare da ƙarfafa iyakar tazara ta biyu ta Watt ba, ɗaya babban shigarwar.

Har yanzu nesa da 1/4

Sabon mai nunawa, 0.3119, har yanzu yana nesa da 0.25 da ake hasashe. Takardar kuma ta bincika ɗaya gefen matsalar — har yaya aka san kuskuren yana girma — inda sakamakon Soundararajan, Lamzouri da wasu suke ƙara bayyana manyan jujjuyawarsa daidai.

Aikin preprint ne mai shafuka 34. Marubutan sun bayyana cewa sun yi amfani da ChatGPT da Codex don tattauna dabarun hujja, taimakawa wajen rubuta da duba sassan hujjoji da tsara manazarta, kuma cewa kowace sanarwa da hujja su ne suka rubuta, suka duba suka kuma tabbatar da ita. Ga ragar ɗigo ta Gauss, yanzu an san cewa kuskuren ba ya girma da sauri fiye da kusan X^{0.312} — da sauran wuri, a ka’ida, har zuwa X^{0.25}.