Harin AI kan tarin lambobi marasa jerin ci gaba: kundin abubuwan da suka makale

TL;DR. Wani mataimakin AI ya yi aiki a kan matsaloli guda biyu game da tarin lambobi marasa jerin ci gaba: hujja don matsalar jimlar juyi (reciprocal-sum), da kuma iyakar quasipolynomial ga kowane tsayin jerin ci gaba...

Mahimman Bayanai

Ayyuka biyu, kuma daya daga cikinsu ne kawai aka taba ikirarin samu. Bayanan sun kasu kashi biyu na ayyuka, kuma yana da kyau a raba matsayinsu daban.
Ragowar yawa ba daya take da damar hada jimla ba. Shinge na farko yana bayyana a matakin tsarin ginin hujjar, kafin a kai ga kowane binciken Fourier.
Rike tsari, da kuma dalilin da ya sa kalmomi uku ba sa gamayya. Wata mafita da aka ba da shawara ita ce a ci gaba da rike tsarin a kusa.
Daidaita mitoci sau daya maimakon sau da yawa. Daidaita ayyukan mita (frequency functions) bi-da-bi shi ne inda matsalar taruwar asara ta fi cizo.

Abu mafi amfani da hujjar da ba a kammala ba za ta iya gaya maka shi ne inda take karyewa takamaimai, kuma wannan tana karyewa a kusan wurare goma sha biyu na musamman.

Takaitaccen Bayani na Zartarwa

Aikin yana da bangarori biyu. Na farko yana gina hujja don matsalar jimlar juyi. Na biyun yana neman iyakar quasipolynomial ga kowane kayyadadden tsayin jerin ci gaba, kuma ba a cimma wannan burin ba.

Tushen farawa shi ne sakamakon jimlar juyi mai kalmomi uku na Bloom da Sisask. Ragowar yawa (density decay) ta fuskar inganci kadai ba ta samar da damar hada jimla (summability) da hujjar ke bukata.

An gwada hanyoyi da dama: tsararrun tarin kewayawa (ambient sets), daidaita mitoci a lokaci guda, sparse counting lemmas, lausasa na Croot-Sisask, dagawa da maimaitawa, kusurwoyi, da entropy a ma'aunin p = log M. Kowannensu ya gamu da takamaiman shinge.

An bar lissafin karshe ba tare da kammalawa ba, babu restriction lemma ko ka'idar kirga mai rarrabe.

Ga wadanda suka kafa kamfanoni da shugabannin fasaha, darasin yana kan tsarin aiki ne: kundin da ke lissafa gazawa da sunayensu ya fi amfani fiye da zane mai cike da karfin gwiwa.

Ayyuka biyu, kuma daya daga cikinsu ne kawai aka taba ikirarin samu #

Bayanan sun kasu kashi biyu na ayyuka, kuma yana da kyau a raba matsayinsu daban. Na farko yana gina hujja don matsalar jimlar juyi. Na biyun yana neman iyakar quasipolynomial ga kowane kayyadadden tsayin jerin ci gaba. Kalmomin aiki suna da muhimmanci. An bayyana aikin farko a matsayin gina hujja. Na biyun yana neman iyaka ne kawai. Babu wani abu a cikin rikodin da ya ce an kai ga buri na biyu, don haka zan dauke shi a matsayin abin da ake hari a duk tsawon wannan rubutu.

Mataimakin ba ya farawa daga babu komai. Yana amfani da sakamakon jimlar juyi mai kalmomi uku na Bloom da Sisask a matsayin tushe. Duk abin da ya biyo baya kokari ne na wuce wannan tushen, ko dai zuwa ga jerin ci gaba masu tsawo ko kuma zuwa ga bayani mai karfi na kididdiga. An tsayar da alamomin lissafi tun da wuri. Ga kowane k, r_k(N) shi ne girman mafi girman karamin tari na {1, ..., N} wanda ba shi da jerin ci gaba mai kalmomi k mai ma'ana. Wani tari A mai yawan α = |A|/N shi ne abin da ake nazari, kuma aikin da aka daidaita (normalised function) shi ne f = 1_A/α.

Wannan daidaitawar ce ke jan ragamar yawancin abubuwan da ke biyo baya. A bayyana μ = 1_A/α, ta yadda E[μ] = 1, sannan a saita M = α^-1. Rashin cikar logarithmic (logarithmic sparsity) a nan shi ne p = log(1/α), wanda yake daidai da log M. Yayin da tarin ke kara rarrabewa (sparse), M da p suna kara girma. Kowane shinge a kasa, ta wata siga ko wata, bayani ne game da abin da ke faruwa yayin da wannan rashin cikar ke karuwa.

Ragowar yawa ba daya take da damar hada jimla ba #

Shinge na farko yana bayyana a matakin tsarin ginin hujjar, kafin a kai ga kowane binciken Fourier. Rikodin ya fada karara cewa ragowar yawa ta fuskar inganci kadai ba ta samar da wannan damar hada jimla (summability). Sanin cewa yawa yana raguwa yayin da kake motsawa a cikin ma'auni bai isa ba. Matsalar jimlar juyi tana bukatar auna ragowar sosai yadda za a iya hada jimlarta.

Wata hanyar tsira mai jan hankali ita ce recurrence (maimaitawar lissafi). Bayanan sun ce recurrence mai karfi sosai don iyakar yawa marasa jerin ci gaba na iya haifar da damar hada jimla a cikin ma'auni masu ninkawa (exponential scales). Wannan jimlar an fada ta cikin taka-tsantsan, kuma ya kamata ta ci gaba da kasancewa haka. Tana bayyana abin da wani recurrence mai karfi na zato zai iya yi. Ba ta bada rahoton cewa an samu daya ba.

Tsada ta biyu daban take da ta farko. Mataimakin ya gano taruwar kananan dangantaka (weak correlations) da yawa a matsayin karin tsada. Kowace dangantaka daya na iya zama marar illa yayin da jimlarsu ke da illa. Wannan batu yana maimaituwa: matakin da yake da arha idan aka yi shi sau daya yana zama mai tsada idan dole ne a maimaita shi.

Rike tsari, da kuma dalilin da ya sa kalmomi uku ba sa gamayya #

Wata mafita da aka ba da shawara ita ce a ci gaba da rike tsarin a kusa. Mataimakin ya ba da shawarar rike tsararrun tarin kewayawa (structured ambient sets) ta hanyar karin matakai masu zuwa. Tunanin, kamar yadda aka fada cikin taka-tsantsan a tushen bayanin, shi ne kada ka jefar da tsararren tarin da ka samu a wani mataki yayin da kake matsawa zuwa na gaba.

Yanayin kalmomi uku yana da wata siffa wadda jerin ci gaba masu tsawo ba su da ita. Tsarin positivity mai amfani ga jerin ci gaba masu kalmomi uku ba ya fadada kai tsaye zuwa ga kirga masu tushe (rooted counts) masu tsawo. Wannan shi ne na farko a cikin wurare da dama inda shingen ya kebanta da k wanda ya fi uku girma. Wani bayani na baya ya fadi abu iri daya a dunkule: jerin ci gaba masu tsawo suna kawo karin shingaye na mataki na sama.

Rikodin ya kunshi zanen tensor-product kan ko kayyadaddun ma'aunin daidaito (uniformity norms) na iya sarrafa ayyukan da aka daidaita masu rarrabe. Mataimakin bai kammala zanen ba. Na gwammace in ba da rahoton hakan maimakon in nuna kamar an warware tambayar ta kowace fuska.

Daidaita mitoci sau daya maimakon sau da yawa #

Daidaita ayyukan mita (frequency functions) bi-da-bi shi ne inda matsalar taruwar asara ta fi cizo. Idan kowane daidaitawa yana cin dan wani abu, dogon jerinsu yana cin abu mai yawa. Mataimakin ya ba da shawarar daidaita ayyukan mita da dama a lokaci guda, maimakon tara asara ta hanyar daidaitawa bi-da-bi. Bayanin gefen shafi ya fito fili ya ce: "Babu asarar maimaitawa idan an daidaita dukkan mitoci a lokaci guda." Kamar shawarar da take takaitawa, wannan jimlar tana da sharadi, domin tana bayyana abin da daidaitawa a lokaci guda zai samar idan ya yi aiki.

Bayanin sai ya fadi abin da har yanzu babu shi. Za a bukaci a iya dawo da S_h daga mitoci masu manyan dangantaka, akalla exp(-poly). Wannan ya ci gaba da kasancewa bukata, ba sakamako ba. Dangantakar da ake tsarawa su ne na ninkawar ma'aurata f(x)f(x + h), tare da f = 1_A/α da kuma α = |A|/N.

Ko da an yarda da hakan, mataimakin har yanzu yana bukatar kiyaye dangantakar dangi (relative correlation) da kuma kammala hujjar kirga. Don haka tunanin daidaitawa a lokaci guda yana da siffa mai ban sha'awa, wato rashin asarar maimaitawa, tare da jerin abubuwan da babu wanda ya tabbatar. Wata hanya mai alaka da wannan tana da iyakarta. Datse kololuwar hadin kai da ba kasafai ake samu ba yana aiki ga gungu-gungu na mitoci masu zaman kansu, amma fadada wannan hujjar zuwa ga gungu-gungu masu dogaro da juna ya ci gaba da kasancewa ba tare da warwarewa ba.

Yankuna marasa komai da ba kasafai ake samu ba da kuma tsadar cikawa mai dacewa #

Lokacin da hujjar ta matsa daga lissafin gefen Fourier zuwa ga kirga jerin ci gaba a zahiri, wata matsalar tsari tana bayyana. Mataimakin ya gano wani shinge: iyakokin sama na yawa a kan tsararrun yankuna suna ba da damar samun yankuna marasa komai da ba kasafai ake samu ba. Iyakar sama kan nawa ne daga cikin A ke kwance a cikin wani yanki ba ta cewa komai game da ko akwai wani bangare na A a wurin. Yankin da ba kasafai ake samu ba kuma babu komai a cikinsa zai iya zama a cikin iyalin yankuna wadanda dukkansu suka cika sharadin iyakar.

Wani martani shi ne cikawa mai dacewa (adaptive filling), kuma yana zuwa da nasa lissafin kudin. Cikawa mai dacewa yana gabatar da karin sharuddan yiwuwa wadanda mataimakin bai tabbatar da su ba. Majorant hanya ce ta dabi'a don kawar da miyagun yankuna da ba kasafai ake samu ba. Rubutun da aka nakalto daga mataimakin ya ce majorant a kan dukkan kananan kwayoyin da ba kasafai ake samu ba da wuya ya zama mai yiwuwa a zahiri. Kalaman suna cikin taka-tsantsan, amma suna nuna hanya mai nisa daga gyara mafi sauki na kai tsaye.

Wannan shi ne tsarin da ke bayyana sauran rikodin. Hujja mai arha tana bukatar iyakar kasa (lower bound) a wani wuri, kayan aikin da ake da su suna ba da iyakokin sama, kuma tazarar da ke tsakaninsu tana cike da abubuwa marasa yawa, rarrabe, kuma marasa komai.

Aron injinan kirga mai rarrabe, da gano cewa abin da aka aro bai isa ba #

Mataimakin ya duba littattafan kirga mai rarrabe (sparse-counting literature) da ake da su don neman hanyar ketarewa. Majiyoyi biyu ne suka şekillendir kokarin.

A kan tsarin Conlon-Fox-Zhao densification, mataimakin ya gano cewa ya dogara ne akan kiyasin linear-forms mai gefe biyu. Hakan yana da muhimmanci domin yanayin da ake ciki a nan yana da tsarin iyakar sama ne ba mai gefe biyu ba. Hanyar da aka gina a kan sarrafawa mai gefe biyu ba ta samuwa lokacin da gefe daya kawai aka sani.

Sparse graph counting lemma na Filmus, Hatami, Hosseini, da Kelman ya samar da samfurin don asymmetric grid moments da positive-semidefinite unbalancing. Don haka ya yi aiki a matsayin samfuri, ba a matsayin ka'idar da za a dauka a saka kai tsaye ba. Duk da haka, wannan wahalar ta sake dawowa. Iyakokin sama kadai sun sha cin karo da ramuka masu rarrabe. Wani karin sarkakiya yana sa grid moments su yi wuyar karantawa: manyan centred grid moments na iya nuna nauyin da ya bata (missing mass) maimakon taruwa zuwa sama. Babban moment yana kama da shaidar taruwa, amma kuma ana iya samar da shi ta hanyar nauyin da ba ya nan. Wannan rudanin shi ne matsalar rami mai rarrabe idan aka kalle ta daga gefen moment.

Bayan haka mataimakin ya yi la'akari da kananan asarar karfi (small power losses) a cikin M. Bai sami wata ka'idar kirga mai dacewa ba. Wannan sakamako ne mai tsafta na rashin samu, kuma yana rage fadin filin binciken: ko da an bar hujjar ta rasa karamin karfi na M, ba a sami ka'idar kirga mai dacewa ba.

Lausasa (smoothing) ya yi tsada sosai don a yi amfani da shi #

Lausasa na Croot-Sisask (Croot-Sisask smoothing) hanya ce ta yau da kullun don daidaita tari kafin a yi kirga. Mataimakin ya duba ta kuma ya gano cewa daidaiton da ake bukata ya yi tsada sosai a cikin aikin da aka ba da shawara. Hanyar ba batattu ba ce. Farashinta ne ya wuce abin da hujjar za ta iya biya.

Wannan tsarin tattalin arzikin ne ke tafiyar da sauran hanyoyin. Ga hujjojin zare (fibre arguments), iyakokin zare ba su ninka kansu kai tsaye ba, kuma constant-norm spheres sun toshe raguwa mai karfi da ake sa rai a kan kayyadaddun haruffan lambobi. Tsarin ninkawa yana nuna cewa iyakoki a kan yankuna ya kamata su hadu su zama iyaka a kan daukacin abin, amma a nan ba su yi hakan ba. Kokarin dagawa da maimaitawa har yanzu ya yi asara mai yawa ta fuskar kididdiga.

An duba hanyar kusan-duality (approximate-duality) ta Ben-Sasson da Ron-Zewi a mataki na gaba. Mataimakin ya gano cewa maimaitawar da aka yi tunani a kai tana yin asara mai yawa ta fuskar kididdiga. Wannan hukunci daya ne da kokarin dagawa, wanda aka iso gare shi daga wata nahiya daban, abin da ke nuna cewa matsalar tana cikin ita kanta maimaitawar ne ba a cikin wani gini daya tak ba. Zan rike wannan a matsayin lura game da tsarin, ba wai cikakken bincike da aka tabbatar ba.

Layi daya na kyakkyawan fata, da lissafin da ya karya shi #

Rikodin yana dauke da wani dan lokaci na kamar an samu ci gaba. Rubutun da aka nakalto daga mataimakin ya tabbatar da cewa ana sarrafa sauye-sauyen darajar ninkawa (products degree fluctuations) tare da daidaiton polynomial, har da alamar motsin rai (!). Tabbaci ne mai dauke da taka-tsantsan, wanda aka nakalto kamar yadda mataimakin ya rubuta shi.

Daga nan sai mataimakin ya lura da matsalar da ke tattare da hakan. Fadadawa masu alama (signed expansions) na iya samun cikakken nauyi na kimanin 2^p, wanda ke danne daidaiton polynomial. A tuna cewa p = log M. Daidaiton polynomial a cikin M buri ne mai sauki, yayin da fadadawa mai alama mai nauyin kimanin 2^p ya fi shi girma nesa ba kusa ba ta fuskar exponential. Daidaiton gaskiya ne, amma soke-soken da ya dogara a kai ba zai iya shanye wannan nauyin mai yawa haka ba.

Ina ganin wannan shi ne shigarwa mafi ban darasi a cikin kundin, domin yana nuna cewa binciken kai-da-kai yana aiki. Ikirarin da aka yi tare da alamar motsin rai an sake bude shi, kuma an gano cewa lissafin da ya dogara a kai yana da nakasu. Haka rikodin aiki na gaskiya yake kama. Dokin da gyaran duk suna zaune a cikin takarda daya.

Kusurwoyi da akwatuna ba sa haduwa su zama jerin ci gaba masu tsawo #

Wata dabarar daban ita ce amfani da sanannen sakamako mai karfi a matsayin matakin takawa. Iyakar kusurwoyi ta quasipolynomial (quasipolynomial corners bound) ita ce kayan aikin da ya dace, amma kusurwoyin da suka fito ba su hadu kai tsaye sun zama jerin ci gaba masu tsawo ba. Kusurwa siffa ce mai fuskoki biyu (two-dimensional) yayin da dogon jerin ci gaba yake da fuska daya (one-dimensional), kuma wucewa a tsakaninsu ba ta faruwa da kanta.

Kirgan akwati mai kalmomi hudu (four-term box count) yana gabatar da wahala mai kama da wannan. Positivity na kirgan akwati mai kalmomi hudu bai sarrafa layinsa na karkata (exceptional diagonal) na musamman ba. Kirga mai kyau (positive count) har yanzu yana iya kasancewa karkashin mamayar gurbatattun tsare-tsare, kuma a kan wannan layin karkata ne suke zaune.

Bayanan suna ba da wata hanya mai yiwuwa ta ci gaba, wadda aka fada cikin taka-tsantsan. Cikakkun ma'aunin akwati (complete box norms) suna ba da wata hanya mai yiwuwa kan ko kirgan sama kadai zai iya tilasta kirgan jerin ci gaba na kasa. Wannan tambayar, ko bayani na gefe daya zai iya tilasta iyakar kasa, ita ce wadda daukacin aikin ke ta kewaya. Duk da haka akwai iyaka kan nisan da wannan zai iya kaiwa. Gudunmawar cubic mara kyau (negative cubic contributions) tana toshe fadadawa kai tsaye zuwa ga tsarin jerin ci gaba.

Koma wa ga entropy a ma'aunin p = log M #

Yayin da aka toshe hanyoyin kirga na kai tsaye, binciken ya juya zuwa ga entropy a ma'aunin p = log M,ដោយ yana amfani da dabarar Kelley-Meka ta amfani da high moments. Zabin ma'aunin ba nauku-uku ba ne: p = log M shi ne ainihin rashin cikar logarithmic da aka bayyana tun a farko.

Rubutun da aka nakalto daga mataimakin yana ba da shawarar amfani da entropy maimakon L2 ta hanyar nonnegativity da rarrabuwar mita (frequency dissociativity na Chang). Wannan shawara ce, ba sakamako ba. Dalilin shi ne μ = 1_A/α ba shi da alamar ragi (nonnegative), kuma dissociativity na Chang yana ba da tsari a gefen mita, don haka entropy na iya yin aikin da adadin L2 ba zai iya yi ba.

Don wannan ya yi aiki mataimakin yana bukatar takamaiman shigarwar bincike (analytic input). Mataimakin ya nemi kiyasin takaitawar Fourier (Fourier restriction estimate) ||g_hat||_q = O(1) don g mai iyaka da kayyadadden q > 2, a karkashin iyakokin sama na cikin gida masu dacewa. Wannan buri ne da aka sa a gaba. Ba a tabbatar da shi ba.

Shingen ya bayyana a mataki na gaba. Ciro tsari daga high moments ya rasa nauyi mai yawa. High moments su ne inda hujjoji irin na Kelley-Meka ke samun karfinsu, kuma a nan ma su ne inda hujjar ta zubar da abin da take bukatar rikewa.

Lissafin da ya fi zuwa kusa #

Lissafin karshe ya cancanci a kalle shi daban, domin yana da nasara da kuma gazawa. Ya kai ga cikakkiyar iyakar karshe (full endpoint bound) a karkashin zato na convolution. Wannan sakamako ne na gaskiya, wanda aka bayyana a haka, kuma yana dauke da sharadin "a karkashin zato na convolution". Iyakar tana aiki idan wadannan zaton sun tabbata.

Amma ya kasa ketare matakin exponent-three da hujjar kirga da aka ba da shawara ke bukata. Yankin da aka ciro daga mataimakin ya lura da raunin iyakar l3 daga convolution da cap, kuma ana bukatar exponent mafi kyau sosai (kasa da 3) don cike gurbi (interpolation). A kalmomi masu sauki, lissafin ya sauka daidai a gefen layin da ba daidai ba. Iyaka a exponent uku ba ta isa ba, kuma interpolation yana bukatar wani abu mafi kyau sosai.

Yana da sauki a karanta kusan-samun-nasara irin wannan a matsayin "an kusa kaiwa". Zan kauce wa hakan. Rikodin bai fadi nisan da lissafin yake da shi daga matakin ba, kuma tazarar da take exponent daya tak har yanzu tana iya zama daukacin matsalar. Abin da rikodin ya fada shi ne mataimakin ya bar lissafin ba tare da kammalawa ba, ba tare da samo restriction lemma ko ka'idar kirga mai rarrabe ba. Wadannan bangarori biyu da suka bata su ne bashin da ke kan aikin.

Darajar kundin #

Zai yi sauki a watsar da wannan a matsayin kokarin da ya ci tura. Ina ganin hakan yana karanta shaidar ba daidai ba, saboda wani dalili da ya shafi fiye da ka'idar lambobi (number theory).

Takardar tana da takamaiman bayani. Tana kiran kowane shinge da sunansa a kowace mahada: yankuna marasa komai da ba kasafai ake samu ba, nauyin da ya bata a cikin grid moments, kiyasi mai gefe biyu a cikin Conlon-Fox-Zhao, nauyi mai alama mai ninkawa a cikin p, layin karkata na musamman na kirgan akwati mai kalmomi hudu, gudunmawar cubic mara kyau. Tana rarrabe tsakanin iyakar da aka gwada aka gani da kuma hanyar da ake sa rai. Tana yi wa recurrence don damar hada jimla alama a matsayin abin da zai iya aiki, kuma aiki na biyu a matsayin abin da ake nema. Tana fadin abin da mataimakin bai kammala ba.

Ga wanda ya kafa kamfani ko shugaban fasaha da ke tantance binciken da AI ke taimakawa, wannan shi ne abin da ya kamata a nema. Tsarin da ke dawo da hujja mai karfin gwiwa kuma mai kama da cikakkiya yana da wuyar bincikawa (audit). Tsarin da ke dawo da jerin gazawa masu suna, tare da kusan-nasara daya da aka yi wa alama karara a matsayin wadda ta gaza ketare mataki, ana iya bincika shi layi bayan layi. Wani zai iya daukar matsalar exponent-three, ko matsalar gungu masu dogaro da juna, kuma ya san inda matakin farawa yake.

Akwai kuma wani tsari da ya kamata a fada kai tsaye. Yawancin gazawar su ne gazawa na taruwa: kananan dangantaka da ke taruwa, daidaitawa bi-da-bi da ke linkuwa, maimaitawar da ke asarar abubuwa da yawa, iyakokin zare da ke kasa ninkawa. Wata mafita da aka ba da shawara ta tunkari wannan kai tsaye, tana daidaita mitoci a lokaci guda ta yadda ba za a sami asarar maimaitawa ba. An fada ta cikin taka-tsantsan, kuma sharadinta na farko, wato dawo da S_h daga mitoci masu manyan dangantaka, ba a tabbatar da shi ba. Duk da haka, ita ce shawarar da ke cikin rikodin wadda ta fi mayar da martani kai tsaye ga korafin da aka fi samu.

Saboda haka tambayar da ke bude wadda rikodin ya tayar takaitacciya ce kuma mai iya gwadawa. Shin za a iya samun iyaka mafi kyau sosai fiye da exponent uku a karkashin zato na convolution wanda tuni ya ba da iyakar karshe? Idan haka ne, interpolation yana da abin da zai yi aiki da shi. Idan ba haka ba, hanyar takaitawar Fourier da ka'idar kirga mai rarrabe, wadanda ba a samo ko daya daga cikinsu ba, su ne wuraren da za a duba na gaba.

Tambayoyin da Ake Yawan Yi #

Mene ne matsalolin lissafi guda biyu da mataimakin AI ya bincika?

Mataimakin ya yi aiki a kan ayyuka biyu masu alaka da juna game da tarin lambobi marasa jerin ci gaba: gina hujja don matsalar jimlar juyi (reciprocal-sum problem), da kuma neman iyakar quasipolynomial ga kowane kayyadadden tsayin jerin ci gaba k. Yayin da ya yi amfani da sakamakon jimlar juyi mai kalmomi uku na Bloom da Sisask a matsayin tushe, ba a cimma iyakar quasipolynomial don kowane tsayin jerin ci gaba ba.

Me ya sa iyakokin sama na yawa suka kasa samar da counting lemma da ake bukata?

Iyakokin sama na yawa a kan tsararrun yankuna suna ba da damar samun yankuna marasa komai da ba kasafai ake samu ba, ma'ana iyakar sama kan nawa ne daga cikin tari A ke kwance a cikin wani yanki ba ta tabbatar da cewa akwai wasu abubuwa na A a wurin ba. A cikin kokari da dama, iyakokin sama kadai sun sha cin karo da ramuka masu rarrabe, kuma manyan centred grid moments na iya nuna nauyin da ya bata maimakon taruwa zuwa sama.

A ina lissafin karshe ya makale?

Lissafin karshe ya kai ga cikakkiyar iyakar karshe a karkashin zato na convolution da aka bayyana, amma ya samar da raunin iyakar l3 kawai wanda ya kasa ketare matakin exponent na kasa da uku sosai da ake bukata don interpolation. Mataimakin ya bar lissafin ba tare da kammalawa ba, ba tare da samo Fourier restriction lemma ko ka'idar kirga mai rarrabe ba.

Manazarta #

Thomas F. Bloom and Olof Sisask, (2020). Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions ⧉. arXiv.
Zander Kelley and Raghu Meka, (2023). Strong Bounds for 3-Progressions ⧉. arXiv.
David Conlon, Jacob Fox, and Yufei Zhao, (2014). A relative Szemerédi theorem ⧉. Geometric and Functional Analysis.
Ernie Croot and Olof Sisask, (2010). A probabilistic technique for finding almost-periods of convolutions ⧉. Geometric and Functional Analysis.

Bita ta ƙarshe 2026-11-01.

Sake buga wannan labari

Harin AI kan tarin lambobi marasa jerin ci gaba: kundin abubuwan da suka makale — Sebastien Rousseau

Wani mataimakin AI ya yi aiki a kan matsaloli guda biyu game da tarin lambobi marasa jerin ci gaba: hujja don matsalar jimlar juyi (reciprocal-sum), da kuma iyakar quasipolynomial ga kowane tsayin jerin ci gaba...

An lasisin wannan labari a karkashin Creative Commons Attribution 4.0 International. Sake bugawa na bukatar nuna asalin URL na asali.

Harin AI kan tarin lambobi marasa jerin ci gaba: kundin abubuwan da suka makale — Sebastien Rousseau

Wani mataimakin AI ya yi aiki a kan matsaloli guda biyu game da tarin lambobi marasa jerin ci gaba: hujja don matsalar jimlar juyi (reciprocal-sum), da kuma iyakar quasipolynomial ga kowane tsayin jerin ci gaba...

Originally published at https://sebastienrousseau.com/ha/2026-10-24-harin-ai-kan-tarin-lambobi-marasa/ by Sebastien Rousseau.
Licensed under CC-BY-4.0.

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