1292:
1093:
1036:
1466:
3495:
732:
1287:{\displaystyle \sum _{a\in A}a^{\ell }={\frac {1}{\ell +1}}\cdot n^{\frac {2\ell +1}{2}}+{\mathcal {O}}\left(n^{\frac {8\ell +3}{8}}\right)+{\mathcal {O}}\left(L^{1/2}\cdot n^{\frac {4\ell +1}{4}}\right)}
821:
616:
2090:
497:
1683:
892:
96:
567:
448:
1610:
309:
406:
2577:
2511:
1086:
885:
1824:
1768:
1520:
357:
2116:
2352:
2325:
1396:
2986:
1930:
2372:
2026:
136:
163:
241:
1314:
2169:
1371:
2398:
2293:
2267:
1956:
2597:
2445:
2241:
2221:
2201:
2140:
1994:
1884:
1864:
1844:
1703:
1540:
1391:
1342:
666:
261:
200:
182:
The main problem in the study of Sidon sequences, posed by Sidon, is to find the maximum number of elements that a Sidon sequence can contain, up to some bound
642:
1976:
form a Sidon sequence. Specifically, he asked if the set of fifth powers is a Sidon set. Ruzsa came close to this by showing that there is a real number
2172:
671:
3314:
1547:
3323:
2731:
1621:
748:
1557:
2599:
is a Sidon set. Therefore all Sidon sets must be Golomb rulers. By a similar argument, all Golomb rulers must be Sidon sets.
2171:
is not a polynomial. The statement that the set of fifth powers is a Sidon set is a special case of the later conjecture of
3611:
2865:
2608:
742:
576:
2031:
1031:{\displaystyle a_{m}=m\cdot n^{1/2}+{\mathcal {O}}\left(n^{7/8}\right)+{\mathcal {O}}\left(L^{1/2}\cdot n^{3/4}\right)}
456:
30:
3451:
3590:
Pilatte, Cédric (2023-03-16). "A solution to the Erdős—Sárközy—Sós problem on asymptotic Sidon bases of order 3".
510:
411:
266:
2839:
366:
2516:
2450:
1043:
826:
2447:
is a Sidon set and not a Golomb ruler. Since it is not a Golomb ruler, there must be four members such that
1777:
644:. The structure of dense Sidon sets has a rich literature and classic constructions by Erdős–Turán, Singer,
2401:
3694:
3689:
2424:
1708:
2914:
1478:
314:
2095:
2330:
2298:
2782:
2683:
3025:
1889:
2357:
3665:
3591:
3572:
3533:
3507:
3476:
3204:
2922:
2877:
2820:
2794:
1999:
101:
3402:
3318:
3048:
1771:
1551:
3657:
3525:
3468:
3273:
3224:
3185:
3165:
3146:
3107:
3068:
3006:
2940:
2895:
2812:
2727:
1826:
with the conjectural density but satisfying only the weaker property that there is a constant
1472:
142:
3649:
3564:
3517:
3460:
3421:
3370:
3332:
3249:
3216:
3177:
3138:
3099:
3060:
2998:
2967:
2932:
2887:
2804:
2763:
2737:
2695:
2656:
645:
220:
3435:
3384:
3344:
3296:
1299:
3431:
3380:
3340:
3292:
2741:
2723:
2145:
1973:
1347:
3310:
2637:
2377:
2272:
2246:
1935:
1543:
214:
2715:
2582:
2430:
2405:
2226:
2206:
2186:
2125:
1979:
1869:
1849:
1829:
1688:
1525:
1468:
That is, infinite Sidon sequences are thinner than the densest finite Sidon sequences.
1461:{\displaystyle \liminf _{x\to \infty }{\frac {A(x){\sqrt {\log x}}}{\sqrt {x}}}\leq 1.}
1376:
1327:
651:
246:
185:
176:
3637:
3336:
3002:
2767:
621:
3683:
3653:
3537:
3398:
3358:
3064:
2824:
2633:
1615:
735:
210:
20:
3669:
3576:
3480:
2417:
2119:
2808:
2400:
in 2023 as a preprint, this later one was posed as a problem in a paper of Erdős,
1475:
and Mian observed that the greedy algorithm gives an infinite Sidon sequence with
3521:
2987:"A theorem in finite projective geometry and some applications to number theory"
2959:
1966:
172:
2971:
2960:"On a Problem of Sidon in Additive Number Theory, and on some Related Problems"
2665:
2660:
3464:
3253:
3181:
2891:
2642:"On a problem of Sidon in additive number theory and on some related problems"
1969:
3661:
3529:
3472:
3228:
3189:
3150:
3111:
3087:
3072:
3010:
2944:
2899:
2816:
3103:
738:, "somehow all known constructions of dense Sidon sets involve the primes".
311:. Several years earlier, James Singer had constructed Sidon sequences with
3552:
3406:
3375:
3220:
2641:
3568:
3426:
2840:"Some problems in number theory, combinatorics and combinatorial geometry"
202:. Despite a large body of research, the question has remained unsolved.
2183:
The existence of Sidon sequences that form an asymptotic basis of order
648:, Spence, Hughes and Cilleruelo have established that a dense Sidon set
1962:
453:
In 1994 Erdős offered 500 dollars for a proof or disproof of the bound
3449:
Kiss, S. Z. (2010-07-01). "On Sidon sets which are asymptotic bases".
2684:"A complete annotated bibliography of work related to Sidon sequences"
3596:
3126:
2613:
3142:
3243:
2936:
2927:
2882:
2799:
1324:
Erdős also showed that, for any particular infinite Sidon sequence
3512:
727:{\displaystyle \left|A\right|\geq \left(1-o(1)\right){\sqrt {n}}}
2700:
3494:
Kiss, Sándor Z.; Rozgonyi, Eszter; Sándor, Csaba (2014-12-01).
2870:
Mathematical
Proceedings of the Cambridge Philosophical Society
1088:. This directly gives some useful asymptotic results including
3407:"Additive properties of random sequences of positive integers"
816:{\displaystyle A=\{a_{1},\dots ,a_{\left|A\right|}\}\subset }
1222:
1178:
1070:
974:
938:
872:
2781:
Balogh, József; Füredi, Zoltán; Roy, Souktik (2023-05-28).
1554:
improved this with a construction of a Sidon sequence with
1961:
Erdős further conjectured that there exists a nonconstant
3496:"On Sidon sets which are asymptotic bases of order $ 4$ "
2754:
Linström, Bern (1969). "An inequality for B2-sequences".
3166:"Combinatorial problems in finite fields and Sidon sets"
2203:(meaning that every sufficiently large natural number
175:, who introduced the concept in his investigations of
2585:
2519:
2453:
2433:
2380:
2360:
2333:
2301:
2275:
2249:
2229:
2209:
2189:
2148:
2128:
2098:
2034:
2002:
1982:
1938:
1892:
1872:
1852:
1832:
1780:
1711:
1691:
1685:
exists. Erdős conjectured that an infinite Sidon set
1624:
1560:
1528:
1481:
1399:
1379:
1350:
1330:
1302:
1096:
1046:
895:
829:
751:
674:
654:
624:
618:
where the maximum is taken over all Sidon subsets of
579:
513:
459:
414:
369:
317:
269:
249:
223:
188:
145:
104:
33:
171:; they are named after the Hungarian mathematician
3612:"First-Year Graduate Finds Paradoxical Number Set"
3500:Functiones et Approximatio Commentarii Mathematici
3088:"Solving a linear equation in a set of integers I"
2591:
2571:
2505:
2439:
2392:
2366:
2346:
2319:
2287:
2261:
2235:
2215:
2195:
2163:
2134:
2110:
2084:
2020:
1988:
1950:
1924:
1878:
1858:
1838:
1818:
1762:
1697:
1677:
1604:
1534:
1514:
1460:
1385:
1365:
1336:
1308:
1286:
1080:
1030:
879:
815:
726:
660:
636:
611:{\displaystyle \left|A\right|=\max \left|S\right|}
610:
561:
491:
442:
400:
351:
303:
255:
235:
194:
157:
130:
90:
3636:Erdős, P.; Sárközy, A.; Sós, V. T. (1994-12-31).
3131:Transactions of the American Mathematical Society
2991:Transactions of the American Mathematical Society
2085:{\displaystyle f(x)=x^{5}+\lfloor cx^{4}\rfloor }
3242:Balasubramanian, R.; Dutta, Sayan (2024-09-08),
1401:
1053:
745:and Sayan Dutta shows that if a dense Sidon set
594:
2913:Eberhard, Sean; Manners, Freddie (2023-02-24).
2243:numbers from the sequence) has been proved for
492:{\displaystyle {\sqrt {x}}+o(x^{\varepsilon })}
167:are different. Sidon sequences are also called
3557:Proceedings of the London Mathematical Society
3030:The Journal of the Indian Mathematical Society
1932:. (To be a Sidon sequence would require that
1678:{\displaystyle A(x)>x^{{\sqrt {2}}-1-o(1)}}
98:of natural numbers in which all pairwise sums
91:{\displaystyle A=\{a_{0},a_{1},a_{2},\dots \}}
3638:"On additive properties of general sequences"
2327:(the sum of four terms with one smaller than
8:
2915:"The Apparent Structure of Dense Sidon Sets"
2105:
2099:
2079:
2063:
1813:
1781:
1075:
1056:
798:
758:
556:
532:
85:
40:
3321:(1981), "A dense infinite Sidon sequence",
562:{\displaystyle A\subset :=\{1,2,\dots ,n\}}
443:{\displaystyle {\sqrt {x}}+0.998{\sqrt{x}}}
2964:Journal of the London Mathematical Society
2783:"An Upper Bound on the Size of Sidon Sets"
2179:Sidon sequences which are asymptotic bases
1614:The best lower bound to date was given by
1373:denoting the number of its elements up to
3595:
3511:
3425:
3374:
3053:Journal of Combinatorial Theory, Series A
2926:
2881:
2798:
2699:
2584:
2579:, which contradicts the proposition that
2563:
2550:
2537:
2524:
2518:
2497:
2484:
2471:
2458:
2452:
2432:
2379:
2359:
2338:
2332:
2300:
2274:
2248:
2228:
2208:
2188:
2147:
2127:
2097:
2073:
2054:
2033:
2001:
1981:
1937:
1910:
1897:
1891:
1871:
1851:
1831:
1801:
1788:
1779:
1735:
1731:
1710:
1690:
1645:
1644:
1623:
1605:{\displaystyle A(x)>{\sqrt{x\log x}}.}
1592:
1576:
1559:
1527:
1505:
1500:
1480:
1431:
1416:
1404:
1398:
1378:
1349:
1329:
1301:
1257:
1240:
1236:
1221:
1220:
1191:
1177:
1176:
1151:
1126:
1117:
1101:
1095:
1069:
1045:
1013:
1009:
992:
988:
973:
972:
955:
951:
937:
936:
923:
919:
900:
894:
871:
854:
850:
838:
830:
828:
784:
765:
750:
717:
673:
653:
623:
578:
512:
480:
460:
458:
433:
428:
415:
413:
385:
380:
370:
368:
318:
316:
304:{\displaystyle {\sqrt {x}}+O({\sqrt{x}})}
291:
286:
270:
268:
248:
222:
187:
144:
122:
109:
103:
73:
60:
47:
32:
3026:"An Affine Analogue of Singer's Theorem"
1618:, who proved that a Sidon sequence with
401:{\displaystyle {\sqrt {x}}+{\sqrt{x}}+1}
2919:The Electronic Journal of Combinatorics
2866:"Solving equations in dense Sidon sets"
2625:
2572:{\displaystyle a_{i}+a_{l}=a_{k}+a_{j}}
2506:{\displaystyle a_{i}-a_{j}=a_{k}-a_{l}}
1081:{\displaystyle L=\max\{0,L^{\prime }\}}
880:{\displaystyle |A|=n^{1/2}-L^{\prime }}
3361:(1998), "An infinite Sidon sequence",
1819:{\displaystyle \{a_{0},a_{1},\dots \}}
243:, the number of elements smaller than
2958:Erdös, P.; Turán, P. (October 1941).
2718:(2004), "C9: Packing sums in pairs",
16:Class of sequences of natural numbers
7:
3553:"On Sidon sets and asymptotic bases"
3551:Cilleruelo, Javier (November 2015).
2028:such that the range of the function
1763:{\displaystyle A(x)>x^{1/2-o(1)}}
3245:The $ m$ -th Element of a Sidon Set
2688:Electronic Journal of Combinatorics
1846:such that for every natural number
1774:showed the existence of a sequence
1515:{\displaystyle A(x)>c{\sqrt{x}}}
352:{\displaystyle {\sqrt {x}}(1-o(1))}
2720:Unsolved problems in number theory
2111:{\displaystyle \lfloor \ \rfloor }
1411:
363:. The upper bound was improved to
14:
3324:European Journal of Combinatorics
3164:Cilleruelo, Javier (2012-05-01).
3003:10.1090/S0002-9947-1938-1501951-4
2787:The American Mathematical Monthly
2354:, for arbitrarily small positive
3049:"Direct product difference sets"
3047:Ganley, Michael J (1977-11-01).
2347:{\displaystyle n^{\varepsilon }}
2320:{\displaystyle m=3+\varepsilon }
3125:Hughes, D. R. (November 1955).
2864:Prendiville, Sean (July 2022).
2756:Journal of Combinatorial Theory
263:in a Sidon sequence is at most
3285:Proc. Natl. Acad. Sci. India A
2158:
2152:
2044:
2038:
1755:
1749:
1721:
1715:
1670:
1664:
1634:
1628:
1570:
1564:
1491:
1485:
1428:
1422:
1408:
1360:
1354:
839:
831:
810:
804:
709:
703:
631:
625:
526:
520:
486:
473:
346:
343:
337:
325:
298:
283:
1:
3337:10.1016/s0195-6698(81)80014-5
3203:Ruzsa, Imre Z. (1999-11-01).
2809:10.1080/00029890.2023.2176667
2768:10.1016/S0021-9800(69)80124-9
2412:Relationship to Golomb rulers
2223:can be written as the sum of
2142:is irrational, this function
1925:{\displaystyle a_{i}+a_{j}=n}
3654:10.1016/0012-365X(94)00108-U
3065:10.1016/0097-3165(77)90023-1
2367:{\displaystyle \varepsilon }
2173:Lander, Parkin and Selfridge
743:Ramachandran Balasubramanian
3127:"Planar Division Neo-Rings"
2423:To see this, suppose for a
2092:is a Sidon sequence, where
2021:{\displaystyle 0<c<1}
131:{\displaystyle a_{i}+a_{j}}
3711:
3452:Acta Mathematica Hungarica
3024:Bose, R. C. (1942-06-01).
2416:All finite Sidon sets are
1886:solutions of the equation
3522:10.7169/facm/2014.51.2.10
3465:10.1007/s10474-010-9155-1
3254:10.48550/arXiv.2409.01986
3182:10.1007/s00493-012-2819-4
2892:10.1017/S0305004121000402
1471:For the other direction,
1296:for any positive integer
3363:Journal of Number Theory
3209:Journal of Number Theory
3205:"Erdős and the Integers"
2972:10.1112/jlms/s1-16.4.212
2661:10.1112/jlms/s1-16.4.212
2609:Moser–de Bruijn sequence
1320:Infinite Sidon sequences
3104:10.4064/aa-65-3-259-282
217:proved that, for every
158:{\displaystyle i\leq j}
3376:10.1006/jnth.1997.2192
3221:10.1006/jnth.1999.2395
2985:Singer, James (1938).
2966:. s1-16 (4): 212–215.
2593:
2573:
2507:
2441:
2394:
2368:
2348:
2321:
2289:
2263:
2237:
2217:
2197:
2165:
2136:
2112:
2086:
2022:
1990:
1952:
1926:
1880:
1860:
1840:
1820:
1764:
1699:
1679:
1606:
1536:
1516:
1462:
1387:
1367:
1338:
1310:
1288:
1082:
1032:
881:
817:
728:
662:
638:
612:
563:
493:
444:
402:
353:
305:
257:
237:
236:{\displaystyle x>0}
196:
159:
132:
92:
3427:10.4064/aa-6-1-83-110
3283:sequences of Sidon",
2847:Mathematica Pannonica
2682:O'Bryant, K. (2004),
2594:
2574:
2508:
2442:
2395:
2369:
2349:
2322:
2290:
2264:
2238:
2218:
2198:
2166:
2137:
2113:
2087:
2023:
1991:
1953:
1927:
1881:
1861:
1841:
1821:
1765:
1700:
1680:
1607:
1537:
1517:
1463:
1388:
1368:
1339:
1311:
1309:{\displaystyle \ell }
1289:
1083:
1033:
882:
818:
729:
663:
639:
613:
564:
507:A Sidon subset
494:
445:
403:
354:
306:
258:
238:
197:
160:
133:
93:
3642:Discrete Mathematics
3086:Ruzsa, Imre (1993).
2838:Erdős, Paul (1994).
2726:, pp. 175–180,
2649:J. London Math. Soc.
2583:
2517:
2451:
2431:
2378:
2358:
2331:
2299:
2273:
2247:
2227:
2207:
2187:
2164:{\displaystyle f(x)}
2146:
2126:
2096:
2032:
2000:
1980:
1972:whose values at the
1936:
1890:
1870:
1850:
1830:
1778:
1709:
1689:
1622:
1558:
1526:
1479:
1397:
1377:
1366:{\displaystyle A(x)}
1348:
1328:
1300:
1094:
1044:
893:
827:
749:
672:
652:
622:
577:
511:
457:
412:
367:
315:
267:
247:
221:
186:
143:
102:
31:
3569:10.1112/plms/pdv050
3272:Mian, Abdul Majid;
2393:{\displaystyle m=3}
2288:{\displaystyle m=4}
2262:{\displaystyle m=5}
1951:{\displaystyle k=1}
741:A recent result of
2589:
2569:
2513:. It follows that
2503:
2437:
2420:, and vice versa.
2390:
2364:
2344:
2317:
2285:
2259:
2233:
2213:
2193:
2161:
2132:
2108:
2082:
2018:
1986:
1948:
1922:
1876:
1866:there are at most
1856:
1836:
1816:
1760:
1695:
1675:
1602:
1532:
1512:
1458:
1415:
1383:
1363:
1334:
1306:
1284:
1112:
1078:
1028:
877:
813:
724:
658:
634:
608:
559:
489:
440:
398:
349:
301:
253:
233:
192:
155:
128:
88:
2592:{\displaystyle S}
2440:{\displaystyle S}
2236:{\displaystyle m}
2216:{\displaystyle n}
2196:{\displaystyle m}
2135:{\displaystyle c}
2104:
1989:{\displaystyle c}
1879:{\displaystyle k}
1859:{\displaystyle n}
1839:{\displaystyle k}
1705:exists for which
1698:{\displaystyle A}
1650:
1597:
1535:{\displaystyle x}
1510:
1450:
1449:
1442:
1400:
1386:{\displaystyle x}
1337:{\displaystyle A}
1276:
1210:
1170:
1142:
1097:
734:. As remarked by
722:
661:{\displaystyle A}
465:
438:
420:
390:
375:
323:
296:
275:
256:{\displaystyle x}
195:{\displaystyle x}
3702:
3674:
3673:
3633:
3627:
3626:
3624:
3623:
3608:
3602:
3601:
3599:
3587:
3581:
3580:
3563:(5): 1206–1230.
3548:
3542:
3541:
3515:
3491:
3485:
3484:
3446:
3440:
3438:
3429:
3414:Acta Arithmetica
3411:
3395:
3389:
3387:
3378:
3355:
3349:
3347:
3307:
3301:
3299:
3276:(1944), "On the
3269:
3263:
3262:
3261:
3260:
3239:
3233:
3232:
3200:
3194:
3193:
3161:
3155:
3154:
3122:
3116:
3115:
3092:Acta Arithmetica
3083:
3077:
3076:
3044:
3038:
3037:
3021:
3015:
3014:
2982:
2976:
2975:
2955:
2949:
2948:
2930:
2910:
2904:
2903:
2885:
2861:
2855:
2854:
2844:
2835:
2829:
2828:
2802:
2778:
2772:
2771:
2751:
2745:
2744:
2722:(3rd ed.),
2712:
2706:
2704:
2703:
2679:
2673:
2663:
2646:
2630:
2598:
2596:
2595:
2590:
2578:
2576:
2575:
2570:
2568:
2567:
2555:
2554:
2542:
2541:
2529:
2528:
2512:
2510:
2509:
2504:
2502:
2501:
2489:
2488:
2476:
2475:
2463:
2462:
2446:
2444:
2443:
2438:
2399:
2397:
2396:
2391:
2373:
2371:
2370:
2365:
2353:
2351:
2350:
2345:
2343:
2342:
2326:
2324:
2323:
2318:
2294:
2292:
2291:
2286:
2268:
2266:
2265:
2260:
2242:
2240:
2239:
2234:
2222:
2220:
2219:
2214:
2202:
2200:
2199:
2194:
2170:
2168:
2167:
2162:
2141:
2139:
2138:
2133:
2117:
2115:
2114:
2109:
2102:
2091:
2089:
2088:
2083:
2078:
2077:
2059:
2058:
2027:
2025:
2024:
2019:
1995:
1993:
1992:
1987:
1957:
1955:
1954:
1949:
1931:
1929:
1928:
1923:
1915:
1914:
1902:
1901:
1885:
1883:
1882:
1877:
1865:
1863:
1862:
1857:
1845:
1843:
1842:
1837:
1825:
1823:
1822:
1817:
1806:
1805:
1793:
1792:
1769:
1767:
1766:
1761:
1759:
1758:
1739:
1704:
1702:
1701:
1696:
1684:
1682:
1681:
1676:
1674:
1673:
1651:
1646:
1611:
1609:
1608:
1603:
1598:
1596:
1591:
1577:
1541:
1539:
1538:
1533:
1521:
1519:
1518:
1513:
1511:
1509:
1501:
1467:
1465:
1464:
1459:
1451:
1445:
1444:
1443:
1432:
1417:
1414:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1343:
1341:
1340:
1335:
1315:
1313:
1312:
1307:
1293:
1291:
1290:
1285:
1283:
1279:
1278:
1277:
1272:
1258:
1249:
1248:
1244:
1226:
1225:
1216:
1212:
1211:
1206:
1192:
1182:
1181:
1172:
1171:
1166:
1152:
1143:
1141:
1127:
1122:
1121:
1111:
1087:
1085:
1084:
1079:
1074:
1073:
1037:
1035:
1034:
1029:
1027:
1023:
1022:
1021:
1017:
1001:
1000:
996:
978:
977:
968:
964:
963:
959:
942:
941:
932:
931:
927:
905:
904:
886:
884:
883:
878:
876:
875:
863:
862:
858:
842:
834:
823:has cardinality
822:
820:
819:
814:
797:
796:
795:
770:
769:
733:
731:
730:
725:
723:
718:
716:
712:
685:
667:
665:
664:
659:
643:
641:
640:
637:{\displaystyle }
635:
617:
615:
614:
609:
607:
590:
568:
566:
565:
560:
503:Dense Sidon Sets
498:
496:
495:
490:
485:
484:
466:
461:
449:
447:
446:
441:
439:
437:
429:
421:
416:
407:
405:
404:
399:
391:
389:
381:
376:
371:
359:terms less than
358:
356:
355:
350:
324:
319:
310:
308:
307:
302:
297:
295:
287:
276:
271:
262:
260:
259:
254:
242:
240:
239:
234:
201:
199:
198:
193:
166:
164:
162:
161:
156:
137:
135:
134:
129:
127:
126:
114:
113:
97:
95:
94:
89:
78:
77:
65:
64:
52:
51:
3710:
3709:
3705:
3704:
3703:
3701:
3700:
3699:
3680:
3679:
3678:
3677:
3635:
3634:
3630:
3621:
3619:
3616:Quanta Magazine
3610:
3609:
3605:
3589:
3588:
3584:
3550:
3549:
3545:
3493:
3492:
3488:
3448:
3447:
3443:
3409:
3397:
3396:
3392:
3357:
3356:
3352:
3309:
3308:
3304:
3282:
3271:
3270:
3266:
3258:
3256:
3241:
3240:
3236:
3202:
3201:
3197:
3163:
3162:
3158:
3143:10.2307/1993000
3124:
3123:
3119:
3085:
3084:
3080:
3046:
3045:
3041:
3023:
3022:
3018:
2984:
2983:
2979:
2957:
2956:
2952:
2921:: P1.33–P1.33.
2912:
2911:
2907:
2863:
2862:
2858:
2842:
2837:
2836:
2832:
2780:
2779:
2775:
2753:
2752:
2748:
2734:
2724:Springer-Verlag
2716:Guy, Richard K.
2714:
2713:
2709:
2681:
2680:
2676:
2644:
2632:
2631:
2627:
2622:
2605:
2581:
2580:
2559:
2546:
2533:
2520:
2515:
2514:
2493:
2480:
2467:
2454:
2449:
2448:
2429:
2428:
2414:
2376:
2375:
2356:
2355:
2334:
2329:
2328:
2297:
2296:
2271:
2270:
2245:
2244:
2225:
2224:
2205:
2204:
2185:
2184:
2181:
2144:
2143:
2124:
2123:
2094:
2093:
2069:
2050:
2030:
2029:
1998:
1997:
1978:
1977:
1974:natural numbers
1934:
1933:
1906:
1893:
1888:
1887:
1868:
1867:
1848:
1847:
1828:
1827:
1797:
1784:
1776:
1775:
1727:
1707:
1706:
1687:
1686:
1640:
1620:
1619:
1578:
1556:
1555:
1524:
1523:
1477:
1476:
1418:
1395:
1394:
1375:
1374:
1346:
1345:
1326:
1325:
1322:
1298:
1297:
1259:
1253:
1232:
1231:
1227:
1193:
1187:
1183:
1153:
1147:
1131:
1113:
1092:
1091:
1065:
1042:
1041:
1005:
984:
983:
979:
947:
943:
915:
896:
891:
890:
867:
846:
825:
824:
785:
780:
761:
747:
746:
693:
689:
675:
670:
669:
650:
649:
620:
619:
597:
580:
575:
574:
509:
508:
505:
476:
455:
454:
410:
409:
408:in 1969 and to
365:
364:
313:
312:
265:
264:
245:
244:
219:
218:
208:
184:
183:
141:
140:
138:
118:
105:
100:
99:
69:
56:
43:
29:
28:
17:
12:
11:
5:
3708:
3706:
3698:
3697:
3692:
3682:
3681:
3676:
3675:
3628:
3603:
3582:
3543:
3486:
3441:
3390:
3350:
3302:
3280:
3264:
3234:
3215:(1): 115–163.
3195:
3176:(5): 497–511.
3156:
3117:
3098:(3): 259–282.
3078:
3059:(3): 321–332.
3039:
3016:
2997:(3): 377–385.
2977:
2950:
2937:10.37236/11191
2905:
2856:
2830:
2793:(5): 437–445.
2773:
2762:(2): 211–212.
2746:
2732:
2707:
2674:
2655:(4): 212–215,
2624:
2623:
2621:
2618:
2617:
2616:
2611:
2604:
2601:
2588:
2566:
2562:
2558:
2553:
2549:
2545:
2540:
2536:
2532:
2527:
2523:
2500:
2496:
2492:
2487:
2483:
2479:
2474:
2470:
2466:
2461:
2457:
2436:
2413:
2410:
2389:
2386:
2383:
2374:) in 2015 and
2363:
2341:
2337:
2316:
2313:
2310:
2307:
2304:
2284:
2281:
2278:
2258:
2255:
2252:
2232:
2212:
2192:
2180:
2177:
2160:
2157:
2154:
2151:
2131:
2107:
2101:
2081:
2076:
2072:
2068:
2065:
2062:
2057:
2053:
2049:
2046:
2043:
2040:
2037:
2017:
2014:
2011:
2008:
2005:
1985:
1947:
1944:
1941:
1921:
1918:
1913:
1909:
1905:
1900:
1896:
1875:
1855:
1835:
1815:
1812:
1809:
1804:
1800:
1796:
1791:
1787:
1783:
1770:holds. He and
1757:
1754:
1751:
1748:
1745:
1742:
1738:
1734:
1730:
1726:
1723:
1720:
1717:
1714:
1694:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1649:
1643:
1639:
1636:
1633:
1630:
1627:
1601:
1595:
1590:
1587:
1584:
1581:
1575:
1572:
1569:
1566:
1563:
1531:
1508:
1504:
1499:
1496:
1493:
1490:
1487:
1484:
1457:
1454:
1448:
1441:
1438:
1435:
1430:
1427:
1424:
1421:
1413:
1410:
1407:
1403:
1402:lim inf
1382:
1362:
1359:
1356:
1353:
1333:
1321:
1318:
1305:
1282:
1275:
1271:
1268:
1265:
1262:
1256:
1252:
1247:
1243:
1239:
1235:
1230:
1224:
1219:
1215:
1209:
1205:
1202:
1199:
1196:
1190:
1186:
1180:
1175:
1169:
1165:
1162:
1159:
1156:
1150:
1146:
1140:
1137:
1134:
1130:
1125:
1120:
1116:
1110:
1107:
1104:
1100:
1077:
1072:
1068:
1064:
1061:
1058:
1055:
1052:
1049:
1026:
1020:
1016:
1012:
1008:
1004:
999:
995:
991:
987:
982:
976:
971:
967:
962:
958:
954:
950:
946:
940:
935:
930:
926:
922:
918:
914:
911:
908:
903:
899:
874:
870:
866:
861:
857:
853:
849:
845:
841:
837:
833:
812:
809:
806:
803:
800:
794:
791:
788:
783:
779:
776:
773:
768:
764:
760:
757:
754:
721:
715:
711:
708:
705:
702:
699:
696:
692:
688:
684:
681:
678:
657:
633:
630:
627:
606:
603:
600:
596:
593:
589:
586:
583:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
504:
501:
488:
483:
479:
475:
472:
469:
464:
436:
432:
427:
424:
419:
397:
394:
388:
384:
379:
374:
348:
345:
342:
339:
336:
333:
330:
327:
322:
300:
294:
290:
285:
282:
279:
274:
252:
232:
229:
226:
207:
204:
191:
177:Fourier series
154:
151:
148:
125:
121:
117:
112:
108:
87:
84:
81:
76:
72:
68:
63:
59:
55:
50:
46:
42:
39:
36:
27:is a sequence
25:Sidon sequence
15:
13:
10:
9:
6:
4:
3:
2:
3707:
3696:
3695:Combinatorics
3693:
3691:
3690:Number theory
3688:
3687:
3685:
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3632:
3629:
3617:
3613:
3607:
3604:
3598:
3593:
3586:
3583:
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3547:
3544:
3539:
3535:
3531:
3527:
3523:
3519:
3514:
3509:
3505:
3501:
3497:
3490:
3487:
3482:
3478:
3474:
3470:
3466:
3462:
3458:
3454:
3453:
3445:
3442:
3437:
3433:
3428:
3423:
3419:
3415:
3408:
3404:
3400:
3394:
3391:
3386:
3382:
3377:
3372:
3368:
3364:
3360:
3354:
3351:
3346:
3342:
3338:
3334:
3330:
3326:
3325:
3320:
3319:Szemerédi, E.
3316:
3312:
3306:
3303:
3298:
3294:
3290:
3286:
3279:
3275:
3268:
3265:
3255:
3251:
3247:
3246:
3238:
3235:
3230:
3226:
3222:
3218:
3214:
3210:
3206:
3199:
3196:
3191:
3187:
3183:
3179:
3175:
3171:
3170:Combinatorica
3167:
3160:
3157:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3121:
3118:
3113:
3109:
3105:
3101:
3097:
3093:
3089:
3082:
3079:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3043:
3040:
3035:
3031:
3027:
3020:
3017:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2981:
2978:
2973:
2969:
2965:
2961:
2954:
2951:
2946:
2942:
2938:
2934:
2929:
2924:
2920:
2916:
2909:
2906:
2901:
2897:
2893:
2889:
2884:
2879:
2875:
2871:
2867:
2860:
2857:
2853:(2): 261–269.
2852:
2848:
2841:
2834:
2831:
2826:
2822:
2818:
2814:
2810:
2806:
2801:
2796:
2792:
2788:
2784:
2777:
2774:
2769:
2765:
2761:
2757:
2750:
2747:
2743:
2739:
2735:
2733:0-387-20860-7
2729:
2725:
2721:
2717:
2711:
2708:
2702:
2697:
2693:
2689:
2685:
2678:
2675:
2671:
2667:
2662:
2658:
2654:
2650:
2643:
2639:
2635:
2629:
2626:
2619:
2615:
2612:
2610:
2607:
2606:
2602:
2600:
2586:
2564:
2560:
2556:
2551:
2547:
2543:
2538:
2534:
2530:
2525:
2521:
2498:
2494:
2490:
2485:
2481:
2477:
2472:
2468:
2464:
2459:
2455:
2434:
2426:
2425:contradiction
2421:
2419:
2418:Golomb rulers
2411:
2409:
2407:
2403:
2387:
2384:
2381:
2361:
2339:
2335:
2314:
2311:
2308:
2305:
2302:
2282:
2279:
2276:
2256:
2253:
2250:
2230:
2210:
2190:
2178:
2176:
2174:
2155:
2149:
2129:
2121:
2074:
2070:
2066:
2060:
2055:
2051:
2047:
2041:
2035:
2015:
2012:
2009:
2006:
2003:
1983:
1975:
1971:
1968:
1964:
1959:
1945:
1942:
1939:
1919:
1916:
1911:
1907:
1903:
1898:
1894:
1873:
1853:
1833:
1810:
1807:
1802:
1798:
1794:
1789:
1785:
1773:
1752:
1746:
1743:
1740:
1736:
1732:
1728:
1724:
1718:
1712:
1692:
1667:
1661:
1658:
1655:
1652:
1647:
1641:
1637:
1631:
1625:
1617:
1616:Imre Z. Ruzsa
1612:
1599:
1593:
1588:
1585:
1582:
1579:
1573:
1567:
1561:
1553:
1549:
1545:
1529:
1506:
1502:
1497:
1494:
1488:
1482:
1474:
1469:
1455:
1452:
1446:
1439:
1436:
1433:
1425:
1419:
1405:
1380:
1357:
1351:
1331:
1319:
1317:
1303:
1294:
1280:
1273:
1269:
1266:
1263:
1260:
1254:
1250:
1245:
1241:
1237:
1233:
1228:
1217:
1213:
1207:
1203:
1200:
1197:
1194:
1188:
1184:
1173:
1167:
1163:
1160:
1157:
1154:
1148:
1144:
1138:
1135:
1132:
1128:
1123:
1118:
1114:
1108:
1105:
1102:
1098:
1089:
1066:
1062:
1059:
1050:
1047:
1038:
1024:
1018:
1014:
1010:
1006:
1002:
997:
993:
989:
985:
980:
969:
965:
960:
956:
952:
948:
944:
933:
928:
924:
920:
916:
912:
909:
906:
901:
897:
888:
868:
864:
859:
855:
851:
847:
843:
835:
807:
801:
792:
789:
786:
781:
777:
774:
771:
766:
762:
755:
752:
744:
739:
737:
719:
713:
706:
700:
697:
694:
690:
686:
682:
679:
676:
655:
647:
628:
604:
601:
598:
591:
587:
584:
581:
572:
553:
550:
547:
544:
541:
538:
535:
529:
523:
517:
514:
502:
500:
481:
477:
470:
467:
462:
451:
434:
430:
425:
422:
417:
395:
392:
386:
382:
377:
372:
362:
340:
334:
331:
328:
320:
292:
288:
280:
277:
272:
250:
230:
227:
224:
216:
212:
206:Early results
205:
203:
189:
180:
178:
174:
170:
152:
149:
146:
123:
119:
115:
110:
106:
82:
79:
74:
70:
66:
61:
57:
53:
48:
44:
37:
34:
26:
22:
21:number theory
3648:(1): 75–99.
3645:
3641:
3631:
3620:. Retrieved
3618:. 2023-06-05
3615:
3606:
3597:2303.09659v1
3585:
3560:
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3456:
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3359:Ruzsa, I. Z.
3353:
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2118:denotes the
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3331:(1): 1–11,
2701:10.37236/32
1967:coefficient
173:Simon Sidon
3684:Categories
3622:2023-06-13
3420:: 83–110,
3315:Komlós, J.
3274:Chowla, S.
3259:2024-09-14
3137:(2): 502.
3036:(0): 1–15.
2928:2107.05744
2883:2005.03484
2800:2103.15850
2742:1058.11001
2620:References
2295:in 2014,
1970:polynomial
1522:for every
668:satisfies
569:is called
450:in 2023.
211:Paul Erdős
169:Sidon sets
3662:0012-365X
3538:119121815
3530:0208-6573
3513:1304.5749
3473:1588-2632
3403:Rényi, A.
3399:Erdős, P.
3369:: 63–71,
3311:Ajtai, M.
3229:0022-314X
3190:1439-6912
3151:0002-9947
3112:0065-1036
3073:0097-3165
3011:0002-9947
2945:1077-8926
2900:0305-0041
2825:232417382
2817:0002-9890
2638:Turán, P.
2634:Erdős, P.
2491:−
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2408:in 1994.
2362:ε
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150:≤
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3670:38168554
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2666:Addendum
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2603:See also
3436:0120213
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3477:S2CID
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2122:. As
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