Difference Sets of Null Density Subsets of
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1 dvces Pue Mthetcs Pulshed Ole M ( Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco Receved Decee 9 ; evsed Feu 9 ; ccepted Feu 6 BSTRCT Let D B : BB : B d fo d B B B lsup s postve the B s cosdeed s lge set D B hs oth hgh dest d ch stuctue The set D B D B D B If B Its dffeece set s lso eltvel lge d t s log stdg coectue tht le sets postve uppe dest the hve thetc pogesso of t legth Hee we show the dffeece set ot e susttl; fo thee ests such tht d d D Kewods: Dffeece Set; Dest; -Set Itoducto suset B of hs ull dest f d B l B D B B Pehps the ost poet geel esult ths cse s theoe coectued Edős d poved Rusz [] It sttes tht f d B the l whee D B : BB : B Ou coce hee s to cosde the ull dest susets the fl : These sets c e cosdeed lge sets og ull dest sets To hve copso sets of hghe destes ote tht f d : d hs Rse popet tht s f d s pttoed to ftel sets the t lest oe of the eleets of the ptto les These two popetes eg vt ude tslto d hvg Rse popet hold fo B : d Blsup () s well Howeve f B the D B s -set [] Ths es D B s oth lge d stuctued set Fo stce t s sdetc: thee ests p such tht pd B fo B defto -set tesects the dffeece set of fte tul sequece d ogst stuctues t s IP whch es thee s sequece of tul ues such tht ll of ts fte sus e B [3] Fo lgeess pot of vew e cosdeed et to fo [4] Note tht thee s lso fl : d lsup whch d s clled the uppe Bch dest of We hve tht d s vt ude ts- lto d stsfes Rse popet [5] lso f B the D B s [5] d B hs thetc pogesso of t legth [6] Howeve B ot e -lge Fo stce B : ut B Edös coectued tht eleets of hve thetc pogesso of t legth [7] pott suset of tul ues whch s the set of pe ues les d the coectue ws poved postvel Copght ScRes
2 96 D DSTJERDI M HOSSEINI Gee-To [8] I ode to peset susttl cotst of sets those we wll show tht fo thee s such tht D D D ; the th dffeece set of eve does ot le Thoughout ths ote uless othewse stted tevl we e tevl of teges So fo stce : Refeece Set Hee we toduce d vestgte the popetes of suset of o-egtve teges sutl defed fo ou lte use Let e set of o-egtve teges the popet tht fo Fo let l e the lgest tege stctl less th Set c : fo ll those l such tht lso set c c f d : othewse Defto Let e s ove The set of oegtve teges R l l l c d s clled the efeece set ssocted to The efeece set c lso e see s follows Set : R the 3 l l 3 l whee : Let us ow ee whe R \ fo gve e- logs to : Le Suppose d let R e ts ssocted efeece set Suppose fo suffcetl lge fo soe The R \ Poof Fst ote tht () ples d lso f we set l : l the l f Now let fo But fo l ; () (3) l l I R l (4) : cd The cd R I cd R cd R cd R I I I So esttg whee R (5) e (6) e : I I 3 3 Note tht (3) fo suffcetl lge we hve 3 Now e I I 3 3 e I I 3 3 I I I lso (5) I I I Theefoe Copght ScRes
3 D DSTJERDI M HOSSEINI e e 3 d 3 d B cosdeg (3) ths ples l e lf lf e l l f d d Now the poof s coplete the to test Let the eple fo stsfg the ove le s Fo ths eple lf l Re 3 Sl guets s the poof of the ove le shows tht f ethe (3) does ot hold o f (3) holds ut thee s soe such tht fo ftel (8) the We gve setch of poof fo the R ltte Suppose s defed s (4) The (8) we hve l sup Let I e s ove d set e : I I Usg the 4 l left eqult (5) we hve Now R I e d l d e l e (7) l Fo (3) we hve Ths d the fct tht ples d we e doe 3 M Result e l sup l sup e We wll show tht copg to set of postve uppe Bch dest the dffeece set D of set d d c e ve spse The et theoe gves clss of eples ths popet Theoe Let the thee ests d D such tht s ot I ptcul d D Poof Let d Cosde the followg setc susets of teges: F (9) F fo d Set : I F F F whee : d I s the tevl Hece d s uo of sutevls of o-zeo teges the fst sutevl of legth d ll othes of legth To hve pctue of s cosde whch s pototpe fo othes The e cosdeed s whch R R R = = = Copght ScRes
4 98 D DSTJERDI M HOSSEINI Tht s hs s hlf eleets s whch e shfted to the left ppoptel So R R B Le oe ots tht R \ \ d hece (Lte we wll pove tht So dvegece of wll e \ cosequece of tht s well) Now we cl tht s ot Fst ote tht : The the cl s estlshed otg tht the ove defto fo whch ples does ot tesect D \ : d so t s ot The sets F s d s e defed such w tht D fo To see ths suppose d e two eleets of d The thee ests such tht d lso whee F So d e o equvletl Theefoe To coplete the poof t es to show tht fo ech We led hve poved ths fct \ fo d we wll pove tht fo d sce we e doe So cosde F (9) d let F : F F : F lso set The : F F Now usg the se gu- ets s Le we pove tht veget If we let the I : cd I : cd s d I I I I Let l e s the defto of R ; the whee \ e e : II 3 l 3 So I I 3 5 e I I l f l e II 5 5 l f 5 d l f 5 d Let e the set defed the poof of the ove theoe d let S e sdetc set the lg- Copght ScRes
5 D DSTJERDI M HOSSEINI 99 est gp slle th The the cocluso of the ove theoe pples fo S Ths cocluso lso holds f we let e p whee e s eve tege d p s cesg tege vlued fucto p lso ecll tht the theoe of Rusz [] fo sets of ull dest sttes tht the dffeece set s cosdelel lge th the set tself tht s f db the () holds Howeve eples such s those the ove theoe oe tes to hve dffeece set whch s s sll s possle Ou ppoch ws to hve the thetc pogesso of log possle legths Theefoe such eples ot ol do ot cotdct the Edös coectue ut stogl e the fvo of t REFERENCES [] I Z Rusz O Dffeece-Sequeces ct thetc Vol pp 5-57 [] V Begelso d N Hd ddtve d Multplctve Rse Theoes -Soe Eleet Results Cotocs Polt d Coputg Vol 993 pp -4 do:7/s [3] V Begelso Ptto Regul Stuctues Coted Lge Sets e udt Joul of Cotol Theo Sees Vol 93 No pp 8-36 do:6/ct36 [4] T C Bow d R Feed thetc Pogessos Lcu Sets Roc Mout Joul of Mthetcs Vol 7 No pp do:6/rmj [5] V Begelso N Hd d R McCutche Notos of Sze d Cotol Popetes of Quotet Sets Segoups Topolog Poceedgs Vol pp 3-6 [6] E Szeed O Sets of Iteges Cotg No Eleets thetc Pogesso ct thetc Vol pp [7] P Edös Poles d Results Cotol Nue Theo stsque 975 pp 95-3 [8] B Gee d T To The Pes Cot tl Log thetc Pogessos ls of Mthetcs Vol 67 No 4 pp do:47/ls86748 Copght ScRes
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